Set-Valued Stochastic Integrals and Applications [1st ed.] 9783030403287, 9783030403294

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Springer Optimization and Its Applications 157

Michał Kisielewicz

Set-Valued Stochastic Integrals and Applications

Springer Optimization and Its Applications Volume 157 Series Editors Panos M. Pardalos (University of Florida) My T. Thai (University of Florida) Honorary Editor Ding-Zhu Du, University of Texas at Dallas Advisory Editors J. Birge (University of Chicago) S. Butenko (Texas A&M University) F. Giannessi (University of Pisa) S. Rebennack (Karlsruhe Institute of Technology) T. Terlaky (Lehigh University) Y. Ye (Stanford University) Aims and Scope Optimization has continued to expand in all directions at an astonishing rate. New algorithmic and theoretical techniques are continually developing and the diffusion into other disciplines is proceeding at a rapid pace, with a spot light on machine learning, artificial intelligence, and quantum computing. Our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in areas not limited to applied mathematics, engineering, medicine, economics, computer science, operations research, and other sciences. The series Springer Optimization and Its Applications (SOIA) aims to publish state-of-the-art expository works (monographs, contributed volumes, textbooks, handbooks) that focus on theory, methods, and applications of optimization. Topics covered include, but are not limited to, nonlinear optimization, combinatorial optimization, continuous optimization, stochastic optimization, Bayesian optimization, optimal control, discrete optimization, multi-objective optimization, and more. New to the series portfolio include Works at the intersection of optimization and machine learning, artificial intelligence, and quantum computing. Volumes from this series are indexed by Web of Science, zbMATH, Mathematical Reviews, and SCOPUS.

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Michał Kisielewicz

Set-Valued Stochastic Integrals and Applications

Michał Kisielewicz Faculty of Mathematics University of Zielona Góra Zielona Góra, Poland

ISSN 1931-6828 ISSN 1931-6836 (electronic) Springer Optimization and Its Applications ISBN 978-3-030-40328-7 ISBN 978-3-030-40329-4 (eBook) https://doi.org/10.1007/978-3-030-40329-4 Mathematics Subject Classification: 28B20, 49J21, 54C60, 65M75, 97K60. © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To the memory of my teacher Professor Andrzej Alexiewicz

Preface

The definition of set-valued integrals extending the classical Minkowski sum of sets, was first suggested by R. Aumann who defined (see [5]) the set-valued integral n of a measurable multifunction F : T → 2R as the image of its subtrajectory integrals S(F ) = {f ∈ Lp (T , F, μ, Rn ) : f (t) ∈ F (t) f or μ − a.e.  t ∈ T } by the linear mapping J : Lp (T , F, μ, Rn ) → Rn defined by J (f ) = T f (t)μ(dt) for every f ∈ Lp (T , F, μ, Rn ). Later on, the above definition has been extended to a more general case (see [4, 24, 25, 27]) dealing with multifunctions with values in the space 2X of all nonempty subsets of a separable Banach space (X, | · |). It is clear that the above definition can be extended to the case where the linear mapping J is defined by a stochastic integral and takes its values from the Banach space L2 (, F, P , X) with a given probability space (, F, P ) and a Hilbert space X. Such approach has been applied by F. Hiai and M. Kisielewicz (see [23, 39]) to the definition of set-valued stochastic functional integrals. In particular, in [39] the linear mapping J, taking its values from the space L2 (, F, Rn ), has been defined by Lebesgue and Itô integrals on the space L2 (R+ × , F , Rn ) of all square integrable F-non-anticipative n-dimensional stochastic processes. Set-valued stochastic functional integrals are good enough (see [42, 44–48]) to the theory of stochastic functional inclusions xt − xs ∈ cl{Js,t (SF (F ◦ x)) + Js,t (SF (G ◦ x))}, called in the author’s monograph [48] as stochastic differential inclusions. Such integrals are not applicable to the theory of stochastic differential inclusions and set-valued stochastic differential equations considered in this book, because these inclusions and equations are defined by set-valued stochastic integrals that have to be set-valued random variables. Therefore, we define set-valued stochastic integrals similarly as it was done in the paper [33]. It is easy to define them for subsets of the space L2 (R+ × , β+ ⊗ F, Rn ) with respect to stochastic processes with paths of bounded variation, because in such a case images of subsets of this space defined by appropriate linear mappings are decomposable subsets of the space L2 (, F, Rn ). Unfortunately, it cannot be applied to set-valued stochastic integrals defined for subsets of the space L2 (R+ × , F , Rn ) with respect to stochastic processes both with bounded and unbounded variation paths. The problem has been partially solved in [33] by E.J. Jung and J.H. Kim. Unfortunately, the set-valued integral vii

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Preface

defined in [33] is still not applicable in the theory of set-valued stochastic differential equations, because the set-valued stochastic integral, defined in [33], is not (see [51, 74] and [54]) integrably bounded. The book is devoted to the general theory of set-valued stochastic integrals, treated as set-valued random variables and defined by images of subsets of the spaces L2 (R+ × , β+ ⊗ F, Rd ) and L2 (R+ × , F , Rd×m ) by linear operators defined by both Lebesgue and Itô integrals. Such defined set-valued stochastic integrals possess properties needed in the theories of stochastic differential inclusions and set-valued stochastic differential equations. Therefore, the main part of applications of such defined set-valued stochastic integrals deals with stochastic differential inclusions and set-valued stochastic differential equations and some applications of such inclusions in the stochastic optimal control theory and in the finance mathematics. Set-valued stochastic integrals presented in the book are connected with set-valued functional integrals considered in the author’s monograph [48]. The content of the book is divided into nine parts. The first three are devoted to the basic notions and theorems of the set theory, the functional analysis, the theory of stochastic processes, multifunctions, and the theory of decomposable subsets of the space Lp (T , F, μ, X). Chapters 4 and 5 are devoted to Aumann, Lebesgue, and Itô set-valued stochastic integrals. The next two chapters present some applications of the above set-valued stochastic integrals to the theories of stochastic differential inclusions, set-valued stochastic differential equations, and set-valued functional inclusions. Chapters 8 and 9 contain some examples of applications of set-valued stochastic integrals to the stochastic optimal control theory and the financial mathematics, respectively. The present book is intended for students, professionals in mathematics, and those interested in applications of the theory of set-valued stochastic integrals. Selected functional analysis and probabilistic methods and the theory of multifunctions are needed for understanding the text. Formulas, theorems, lemmas, remarks, and corollaries are numbered separately in each chapter and denoted by three numbers. The first stands for the chapter number, the second for the number of the section, and the last for the number formula, theorem, etc. The ends of proofs, theorems, remarks, and corollaries are denoted by . The main information on bibliographical sources of the material presented in each chapter are contained in the last part of the chapter entitled Notes and Remarks. The manuscript of this book was read by my colleagues M. Michta and J. Motyl who made many valuable comments. The last version of the manuscript was read by Professor Diethard Pallaschke. His remarks were very useful in my last correction of the manuscript. It is my pleasure to thank all of them for their efforts. Zielona Góra, Poland

Michał Kisielewicz

Contents

1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Set Theory and Topological Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Functional Analysis Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Space of Subsets of Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Lebesgue and Bochner Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Properties of Exit Times of Continuous Processes . . . . . . . . . . . . . . . . . . . . 1.8 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 12 17 24 37 42 43 58

2

Multifunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Continuity of Multifunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Measurability of Multifunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Subtrajectory Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 69 73 79

3

Decomposable Subsets of Lp (T , F , µ, X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.1 The Space Lp (T , F, μ, X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.2 Decomposable Subsets of Lp (T , F, μ, X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.3 Decomposable Hulls of Subsets of Lp (T , F, μ, X) . . . . . . . . . . . . . . . . . . 91 3.4 Conditional Expectation of Subsets of Lp (T , F, μ, X) . . . . . . . . . . . . . . 100 3.5 Set-Valued Martingales and Martingale Selectors . . . . . . . . . . . . . . . . . . . . . 103 3.6 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4

Aumann Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Aumann Integrals of Subsets of Lp (T , F, μ, X). . . . . . . . . . . . . . . . . . . . . . 4.2 Aumann Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Lebesgue Set-Valued Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Approximation of Aumann Stochastic Integrals. . . . . . . . . . . . . . . . . . . . . . . 4.5 Selection Theorems for Aumann Stochastic Integrals . . . . . . . . . . . . . . . . 4.6 Indefinite Aumann Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 117 121 122 134 137 138 ix

x

Contents

5

Itô Set-Valued Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Itô Set-Valued Functional Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Itô Set-Valued Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Unboundedness of Itô Set-Valued Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Boundedness of Itô Set-Valued Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Indefinite Itô Set-Valued Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Integral Representation of Set-Valued Martingales . . . . . . . . . . . . . . . . . . . 5.7 Approximation of Itô Set-Valued Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Selection Theorems for Itô Set-Valued Integrals . . . . . . . . . . . . . . . . . . . . . . 5.9 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 141 150 155 163 167 172 174 187 193

6

Stochastic Differential Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Existence of Solutions of SDI (F, G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Properties of Strong Solutions Sets of SDI (F, G) . . . . . . . . . . . . . . . . . . . . 6.3 Weak Compactness of Weak Solutions Sets of SDI (F, G). . . . . . . . . . . 6.4 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 195 198 200 209

7

Set-Valued Stochastic Equations and Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Existence of Strong Solutions of SDE(F, G) . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Existence of Weak Solutions of SDE(F, G) . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Weak Compactness of Weak Solutions Sets of SDE(F, G) . . . . . . . . . . 7.4 Set-Valued Functional Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Attainable Sets of Stochastic Functional Inclusions . . . . . . . . . . . . . . . . . . 7.6 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

211 211 217 225 232 237 247

8

Stochastic Optimal Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Optimal Control Problems for Systems Described by SDE(f, g) . . . 8.2 Optimal Control Problems for Systems Described by SDI (F, G) . . . 8.3 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249 249 255 257

9

Mathematical Finance Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Market, Portfolio, and Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Option Pricing and Consumption Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Finance Optimal Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Recursive Utility Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259 259 261 262 270 273

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

List of Symbols

N R ∅ a∈A A⊂B A\B lim sup An lim inf An (X, T ) cl[S], S (X, ρ) (X,  · ) X∗ σ (X, Y ) σ (X, X∗ ) s(·, A) co S co S σ (X∗ , X) Cl(X) Comp(X) Conv(X) ¯ h(A, B) h(A, B) P(X) Li An Ls An Lim An A B RNP Pn ⇒ P

Set of positive integers, 1 Set of real numbers, 1 Empty set, 1 a is an element of a set A, 1 A is a subset of B (set inclusion), 1 Complement of B with respect to A, 1 Upper limit of a sequence of sets, 1 Lower limit of a sequence of sets, 1 Topological space, 2 Closure of a set S of (X,T ), 2 Metric space, 4 Normed space, 5 Normed conjugate (dual) of X, 7 Weak topology with respect to duality, 7 Weak topology of a normed space X, 7 Support function of a set A, 7 Convex hull of a set S, 8 Closed convex hull of a set S, 8 Weak∗ topology of X∗ , 11 Family of nonempty closed subsets of (X,ρ), 12 Family of nonempty compact subsets of (X,ρ), 12 Family of nonempty compact convex subsets of (X,| · |), 12 Hausdorff subdistance between sets A and B, 12 Hausdorff distance between sets A and B, 12 Space of nonempty subsets of the metric space (X,ρ), 11 Topological lower limit of a set-valued sequence, 16 Topological upper limit of a set-valued sequence, 16 Topological limit of a set-valued sequence, 16 Symmetric difference of sets A and B, 17 Radon–Nikodym property, 23 Weak convergence of sequences of probability measures, 28 xi

xii

List of Symbols

Xn ⇒ X P

Xn → X C(R+ , Rd ) C(Rd ) P(F) O(F) X, Y X B = (Bt )t≥0 D(F, Rd ) Graph(F ) l.s.c. u.s.c. h − l.s.c. h − u.s.c. F − (E) dec K dec K decG K decG K M(T , X) S(F ) A(T , X) CF (T , Rd )

Convergence in distribution of random variables, 30 Convergence in probability of random variables, 30 Space of continuous functions f : R+ → Rd , 31 Space C(R+ , Rd ), 31 Predictable σ algebra on R+ × F, 37 Optional σ algebra on R+ × F, 37 Cross variation of X and Y, 40 Quadratic variation of X, 40 Brownian motion, 40 The space of F cadlag stochastic processes, 42 Graph of multifunction F, 61 Lower semicontinuous, 61 Upper semicontinuous, 61 h-lower semicontinuous, 62 h-upper semicontinuous, 62 Subset of T defined by {t ∈ T : F(t) ∩ E = ∅}, 69 Decomposable hull of a set K, 73 Closed decomposable hull of a set K, 73 G-Decomposable hull of a set K, 73 G-Closed decomposable hull of a set K, 73 Family of all measurable functions f:T→X, 73 Subtrajectory integrals of multifunctions F : T → Cl(X), 74 Family of multifunctions F : T → Cl(X) such that S(F) = ∅, 74 Space of continuous F-adapted stochastic processes, 198

Chapter 1

Preliminaries

In this chapter we present a survey of concepts and results of the fields of set theory, topology, functional analysis and theory of stochastic processes, that are used in the book. The greater part of all results is stated without proofs which can be found in the standard monographs. It is assumed that the basic notions of measure and probability theories are known to the reader.

1.1 Set Theory and Topological Preliminaries The sets of positive integers and real numbers are denoted by N and R, respectively. Capital Latin or Greek are usually used to denote sets, collections, families, or classes. The symbol ∈ will indicate membership in a set. The void (empty) set is denoted by ∅. If every element of a set A is also an element of a set B, then we write A ⊂ B. For given sets A and B, by A ∪ B, A ∩ B, and A \ B the union, the intersection, and the complement of B to A are denoted, respectively. Given sets A and B, the equality A = B is written if and only if A ⊂ B and B ⊂ A. They are called disjoint if A ∩ B  = ∅. If M is a family of sets, i.e., a set whose elements are sets, then M or {A : A ∈ M} denotes the union of all   elements of M. Similarly, M or {A : A ∈ M} denotes the intersection of all members of M. if M = {Aλ : λ ∈ }, In particular,  where   and Aλ are given sets, we write λ∈ Aλ and λ∈ Aλ , instead of M and M, respectively. ∞ n  If  = N or  = {1, 2, . . . , n} we write A or A and ∞ n k n=1 k=1 n=1 An or n   λ∈ Aλ and λ∈ Aλ , respectively. A family of sets has the k=1 Ak instead of finite intersection property if its every finite subfamily has a non-void intersection.  ∞ Finally, if M  = {An: n ∈ N} we define lim sup An = : ∞ n=1 k=n Ak and ∞ ∞ i : i, n = lim inf An = : n=1 A . It can be verified that for every family {A k n k=n ∞ ∞ i i 1, 2, . . .} one has i=1 [lim inf An ] = lim inf[ i=1 An ]. If lim sup An = lim inf An , © Springer Nature Switzerland AG 2020 M. Kisielewicz, Set-Valued Stochastic Integrals and Applications, Springer Optimization and Its Applications 157, https://doi.org/10.1007/978-3-030-40329-4_1

1

2

1 Preliminaries

then we say that a sequence (An )∞ n=1 converges. Given nonempty sets X and Y , a function f defined on a set X with values at Y is denoted by f : X → Y and its value at a point x ∈ X is denoted by f (x). In particular, if X = N, then a function f : N → Y is said to be a sequence of points yn = f (n) of a set Y . It is denoted ∞ by (yn ) or (yn )∞ n=1 . It is clear that the sequence (yn )n=1 is not a subset of the space ∞ Y . But for simplicity we shall write (yn )n=1 ⊂ Y instead of to say that (yn )∞ n=1 is a sequence of points of the set Y . From the axiomatic viewpoint, a topology T in a set X is a family of subsets of X satisfying: the union of any collection of sets in T is again in T , the intersection of a finite collection of sets of T is a set in T and such that X itself and the empty set ∅ belong to T . The elements of T are called open sets and the set X with its topology is called a topological space. It is written as a pair (X, T ). A base for the topology T is a collection B of open subsets of X such that any element of T can be written as a union of elements from B. A nonempty collection S of open subsets of X is called a sub-base for the topology T if the collection of all finite intersections of elements of S is a base for T . Closed sets are defined as the complements with respect to X of open  sets. It is clear that for a collection  of closed subsets of X its intersection  is a closed subset of X. Given a subset S of a topological space (X, T ) its closure, denoted by cl[S] or S, is defined to be the intersection of all closed subsets of X containing S. It is clear that S ⊂ S. The union of all open subsets of S is denoted by Int(S) and said to be the interior of S. It can be verified that S is closed if and only if S = S, which implies that X \ S = Int(S). Any open set of a topological space (X, T ) which contains a point x ∈ X is called a neighborhood of x. Similarly, any open set which contains a set S ⊂ X is called a neighborhood of S. If (X, T ) is a topological space with the property that each pair of distinct points of X has disjoint neighborhoods, then its topology T is said to be a Hausdorff topology, and (X, T ) is called a Hausdorff topological space or simply Hausdorff space. If we have two topologies T1 and T2 in a set X, the topology T1 is said to be weaker, then the topology T2 if T1 ⊂ T2 . Evidently, in this case T2 is said to be stronger than T1 . The above topologies are the same or equivalent if T1 = T2 . A sequence (xn ) of points xn of a topological space (X, T ) is said to be convergent with respect to topology T to a limit x0 ∈ X (written xn → x0 or lim xn = x0 ) if each neighborhood of x0 contains all but a finite numbers of elements of (xn ). In the general case a convergent sequence can possess a more than one limit. In the Hausdorff space every convergent sequence possesses exactly one limit. Let S be a subset of a topological space (X, T ). The set S is said to be sequentially closed if for every convergent sequence of points of S its limit belongs to S. By the sequentially closure of S we mean the set of limits of all convergent sequences of points of S. This is denoted by Seqcl(S). It is clear that S ⊂ Seqcl(S) ⊂ S and therefore, every closed set is sequentially closed. A set S is said to be dense in X if S = X. A set S ⊂ X is said to be connected if do not exist subsets P and Q of X such that P ∩ Q = P ∩ Q = ∅ and S = P ∪ Q. Let (X, T ) be a topological space and let S be a subset of X. Then we can define a topology for S to be the collection of sets of the form {U ∩S : U ∈ T }. This topology is called the relative topology on S generated by the topology T . A topological space

1.1 Set Theory and Topological Preliminaries

3

(X, T ) is called separable if there exists a countable dense subset S in X, i.e., such that X = S. It is clear that if the topological space (X, T ) is separable, then for every nonempty set S ⊂ X, a topological space S with its relative topology generated by T is separable. A family β of open sets in X is said to be an open covering of S if every point of S belongs to at least one element of β. A set S is called compact if every open covering of S contains a finite subfamily which covers S. S is said to be sequentially compact if every sequence in S has a subsequence which converges to a point of S. A set S is called relatively compact if its closure S is compact. S is called relatively sequentially compact if every sequence of S has a subsequence which converges to a point in X. A topological space is called compact (sequentially compact) if the set X is compact (sequentially compact). It is clear that every closed (sequentially closed) subset of a compact (sequentially compact) topological space is compact (sequentially compact). A topological space (X, T ) is said to be perfect separable if there is a countable basis for its topology. If X is a nonempty topological space, it may be possible in various ways to make a new space Y which is compact, which contains X as a dense subset of Y , and which is such that the original topology in X is identical with the relative topology of X generated by the topology of Y . A space Y related to X in this way is called a compactification of X. In particular, it can be proved that a perfectly separable noncompact topological space has a perfectly separable compactification. Let {Ui }i∈I and {Vj }ij∈J be two coverings of a topological space (X, T ). A covering {Ui }i∈I is said to be a refinement of {Vj }ij∈J if for every i ∈ I there exists ji ∈ J such that Ui ⊂ Vji . A covering {Ui }i∈I is called locally finite if for every x ∈ X there exists a neighborhood V of x such that Ui ∩ V = ∅ only for finite of indexes i ∈ I . A topological space (X, T ) is called paracompact if it is a Hausdorff space and its each open covering has a locally finite open refinement. Let (X, TX ) and (Y, TY ) be topological spaces and f : X → Y be a function. Then f is said to be continuous at a point x0 ∈ X if to each neighborhood V ∈ TY of f (x0 ) there is a neighborhood U ∈ TX of x0 such that f (U ) ⊂ V . A function f : X → Y is said to be sequentially continuous at a point x0 ∈ X if for every sequence (xn ) of X convergent to x0 the sequence (f (xn )) converges to f (x0 ). A function f : X → Y is called continuous (sequentially continuous) if it is continuous (sequentially continuous) at each point x0 ∈ X. It can be easily verify that for every compact (sequentially compact) set S ⊂ X its image f (S) by continuous (sequentially continuous) function f : X → Y is a compact (sequentialy compact) subset of the space Y . If f : X → Y is continuous, then for every set A ⊂ X one has f ([A]X ) ⊂ [f (A)]Y , where [A]X and [f (A)]Y denote closures with respect to topologies TX and TY , respectively. A function f : X → R is called a functional. The following result is important in applications. Theorem 1.1.1 A topological space is compact if and only if every family of closed sets with the finite intersection property has a non-void intersection.  

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1.2 Functional Analysis Preliminaries Let X be an arbitrary set and S a given collection of subsets of X. The topology in X generated by S is a unique topology being the intersection of all topologies in X containing S. A constructive way to characterize the unique weakest topology containing S is by taking all unions of finite intersections of elements of S, together with ∅ and X, as topology for X. It has S as its sub-base. There are another ways of the definition of topologies in a given set. As an example, which will be useful later, let X be any set and let (Y, TY ) be a topological space and {fα : α ∈ I } be a collection of functions, each defined on X with range in Y . The topology generated by {fα : α ∈ I } is the weakest topology in X under which the functions fα are continuous. To obtain such topology, let us note that this requires that fα−1 (U ) has to be an open set in X for each α ∈ I and U ∈ TY , where fα−1 (U ) = {x ∈ X : fα (x) ∈ U }. Let S = {fα−1 (U ) : α ∈ I, U ∈ TY } and use S, as in the previous passage, to generate a topology. This topology with S as sub-base is the topology generated by {fα : α ∈ I }. Let X1 , . . . , Xn be topological spaces with Bi as basis for the topology of Xi . Their topological product X1 × . . . × Xn is defined as the set of all n-tuples (x1 , . . . , xn ) with xi ∈ Xi , taking as a base for topology all products U1 ×. . .×Un of Ui ∈ Bi . It can be proved that the topological product of compact spaces is compact. Another very useful way of generating a topology for a set X is via a metric function which is a real-valued function ρ defined on X × X satisfying: ρ(x, y) = ρ(y, x), ρ(x, z) ≤ ρ(x, y) + ρ(y, z) and ρ(x, y) = 0 if and only if x = y for x, y, z ∈ X. It follows that the values of ρ are non-negative. A set X with a metric ρ considered as topological space with a base defined by the open balls B(x, r) = : {y ∈ X : ρ(x, y) < r}, with x ∈ X and r > 0, is called a metric space. It is written as a pair (X, ρ). It is clear that a metric space is the Hausdorff space. From the above definitions it follows that for a given nonempty set S of a metric space we have that x ∈ S if and only if there exists a sequence (xn ) of points of S convergent to x. An important question is: given a set X with a topology T , can elements of T be defined via a metric function? If this is possible the topology T is said to be metrizable. Necessary and sufficient conditions for a topology to be metrizabale can be given. In particular, a compact topological space is metrizable if and only if it is perfect separable. In a metric space (X, ρ), compactness and sequential compactness of a set S are equivalent. Furthermore, it can be proved (see [15], Th. 15 of Chap. I) that S is relatively compact if and only if it is totally bounded,  i.e., if for every ε > 0 there is a finite set {x1 , . . . , xn } ⊂ S such that S ⊂ ni=1 B(xi , ε), where B(xi , ε) = {x ∈ X : ρ(x, xi ) < ε}. A metric space is perfectly separable if and only if it is separable. Hence in particular, it follows that a separable metric space has a metric compactification. Furthermore, it can be proved that for any locally finite open covering of a metric space X we can associate a locally Lipschitzean partition of unity subordinate to it. Recall, a family (ϕi )i∈I of real-valued functions ϕi defined on X is said to be a locally Lipschitzean partition of unity if: (i) ϕi is for all i ∈ I locally Lipschitz and non-negative, (ii) a family

1.2 Functional Analysis Preliminaries

5

{supp ϕi }i∈I with  supp ϕi = {x ∈ X : ϕi (x) = 0} is a closed locally finite covering of X, and (iii) i∈I ϕi (x) = 1 for each x ∈ X. A sequence (xn ) in a metric space (X, ρ) is called a Cauchy sequence if for every ε > 0 there is a positive integer N such that ρ(xn , xm ) < ε whenever n, m ≥ N . A metric space is said to be complete if every Cauchy sequence converges to an element of the space. A complete separable metric space is called a Polish space. A subset S of a metric space X is called bounded if there are x0 ∈ X and M > 0 such that ρ(x0 , x) ≤ M for every x ∈ S. It is clear that every compact subset of a metric space is bounded. It is also clear that every Cauchy sequence (xn ) of a metric space is bounded. The following result is important in the further applications. Theorem 1.2.1 Let (xn,m ) be a sequence of a complete metric space (X, ρ) such that for every m ∈ N the limit g(m) = : limn→∞ xn,m exists and such that the limit h(n) = : limm→∞ xn,m exists uniformly with respect to n ∈ N. Then the limits limn→∞ [limm→∞ xn,m ] and limm→∞ [limn→∞ xn,m ] exist and are equal to the limit limn,m→∞ xn,m .   Let X be a linear space over a field K. Assume X is also a topological space and K is either a real or a complex scalar field with their usual topology. Then X is called a linear topological space if the mappings X × X  (x1 , x2 ) → x1 + x2 ∈ X and K × X  (α, x) → αx ∈ X are continuous. A normed space is a linear space X together with a function · : X → R, called the norm, that satisfies: x = 0 if and only if x = 0, x + y ≤ x + y and αx = |α|x for x, y ∈ X, and α ∈ K. Given a norm  ·  we can define a metric ρ on X × X by setting ρ(x, y) = x − y, which can be used to define a topology on X. Then the norm can be used to define a topology, called simply the norm topology. If a normed space, understood as metric space with a metric defined by its norm, is complete, then it is called a Banach space. From the above definition the following result follows. Lemma 1.2.1 A normed  space (X, ·) is complete ∞ if and only if for every sequence (xn ) of X such that ∞ x  < ∞ a series   n n=1 n=1 xn is convergent. Given subsets A and B of a linear space X and a number α ∈ R we can define sets A+B and α·A by settings A+B = {a+b : a ∈ A, b ∈ B} and α·A = {α·a : a ∈ A}. A set A + B is called the Minkowski sum of sets A and B. Lemma 1.2.2 Let A and B be subsets of a normed space (X,  · ). Then cl(A + B) = cl(A + B), where closures are taken with respect to the norm topology. Proof It is clear that cl(A + B) ⊂ cl(A + B). On the other hand, for every ε > 0 and u ∈ cl(A + B) there are uε ∈ A and vε ∈ B such that u − (uε + vε ) ≤ ε/3. Similarly, we infer that there are aε ∈ A and bε ∈ B such that uε − aε  ≤ ε/3 and vε − bε  ≤ ε/3. Then u − (aε + bε ) ≤ ε, which implies that u ∈ cl(A + B).   Lemma 1.2.3 If A and B are sequentially closed subsets of a linear topological space (X, T ), with B sequentially compact, then A + B is sequentially closed.

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Proof We have A + B ⊂ Seqcl(A + B). Furthermore, for every u ∈ Seqcl(A + B) there are sequences (an ) and (bn ) of A and B, respectively, such that an + bn → u as n → ∞. By sequentially compactness of B there exists a subsequence (bnk ) of (bn ) and b ∈ B such that bnk → b as k → ∞. Then ank + (bnk − u) → 0 and (bnk − u) → b − u as k → ∞. Therefore, ank → u − b as k → ∞, which by sequentially closedness of A implies that u − b ∈ A. Then u = (u − b) + b ∈ A + B for every u ∈ Seqcl(A + B). Thus, Seqcl(A + B) ⊂ A + B   Remark 1.2.1 Similarly to the proof of Lemma 1.2.3 it can be verified that if A and B are closed subsets of a normed space (X,  · ) with B compact, then A + B is closed. Let X be a linear space and K be a set in X. A set K is said to be convex if, whenever x, y ∈ K, the “line segment” λx + (1 − λ)y, 0 ≤ λ ≤ 1 joining x and y also belongs to K. If X and Y are linear spaces, then a mapping T : X → Y is said to be an affine mapping if T [λx + (1 − λ)y] = λT (x) + (1 − λ)T (y) for every x, y ∈ X and λ ∈ [0, 1]. It is clear that if T is the affine mapping from a linear space X to a linear space Y , and K is a convex set in X then T (K) is convex. It is evident that every convex set K in a linear space is connected, because, for every x, y ∈ K it contains a connected subset Lxy = {λx + (1 − λ)y, 0 ≤ λ ≤ 1} such that x, y ∈ Lxy . From the above definition it follows that the intersection of an arbitrary family of convex subsets of a linear space is convex. Furthermore, convex sets possess the following properties. Lemma 1.2.4 Let K be a convex set of a real linear vector space X. For every x1 , . . . , xn ∈ K and every non-negative numbers λ1 , . . . , λn ∈ [0, 1] such that λ1 + . . . + λn = 1 one has λ1 x1 + . . . + λn xn ∈ K.   Lemma 1.2.5 If K, H are convex subsets of a real linear vector space X, then α · K and α · K + β · H are convex for every α, β ∈ R.   Lemma 1.2.6 The closure of a convex set in a linear topological space is convex. Proof Let us observe that a set K of a linear topological space X is convex if and only if the mapping ϕ : X × X × [0, 1]  (x, y, λ) → λx + (1 − λ)y ∈ X sends K × K × [0, 1] into K. But ϕ is continuous and K × K × [0, 1] = K × K × [0, 1]. Then ϕ(K × K × [0, 1]) = ϕ(K × K × [0, 1]) ⊂ ϕ(K × K × [0, 1]) ⊂ K whenever K is convex. Thus, K is convex if K is convex.   If X and Y are normed spaces, then a mapping T : X → Y is said to be linear operator if T (αx + βy) = αT (x) + βT (y) for every x, y ∈ X and α, β ∈ R. It can be verified (see [15], Lemma 3.2 of Chap. II) that a linear operator T : X → Y is continuous if and only if there is M > 0 such that T (x) ≤ Mx for all x ∈ X. The set of all linear continuous operators from X to Y, with the usual definitions of addition and scalar multiplication, is itself a linear space. It will be denoted by L(X, Y ). One may define a norm of an element T ∈ L(X, Y ) by setting | T | = sup{T (x) : x ≤ 1}. An associated | · | -topology is called the uniform operator topology for L(X, Y ).

1.2 Functional Analysis Preliminaries

7

Consider now the space L(X, Y ) when X is a normed space and Y is the scalar field K, which is itself a normed space. In such case the space L(X, Y ) with the norm defined as above is denoted by X∗ and called the conjugate or dual space of X. Elements of X∗ can be used to generate weak topology for X, which is the weakest topology for X under which the elements of X∗ are still continuous. Such defined weak topology of X is denoted by σ (X, X∗ ). It can be verified that the weak topology σ (X, X∗ ) is the Hausdorff topology. A sequence (xn ) in a normed space X is convergent weakly to x (converges in the weak topology) if and only if the scalar sequence {x ∗ (xn − x)} converges to zero for each fixed x ∗ ∈ X∗ . Geometrically this means that the distance (in norm) from xn to each hyperplane through x tends to zero. The point x is called a weak limit of the sequence (xn ), and the sequence (xn ) is said to converge weakly to x. Every sequence (xn ) of X such that {x ∗ (xn )} is a Cauchy sequence of scalars for each x ∗ ∈ X∗ is called a weak Cauchy sequence. It can be easily verified that every weakly convergent sequence (xn ) of a normed space is bounded. It is clear that a sequence (xn ) of a normed space is weakly convergent to x if and only if every sub-sequence of (xn ) has a sub-sequence weakly convergent to x. Given a nonempty set A ⊂ X the support function of A defined on X∗ with values at R = R∪{+∞} is denoted by s(·, A). It is defined by setting s(x ∗ , A) = sup{x ∗ (a) : a ∈ A} for every x ∗ ∈ X∗ . Throughout the book by clw (A) we denoted the closure of a set A ⊂ X with respect to the weak topology of a normed space (X,  · ), whereas cl(A) denotes the closure of a set A ⊂ X with respect to its norm topology. Theorem (Dunford–Schwartz) Let T be a linear mapping of a Banach space X into a Banach space Y . Then T is continuous with respect to the norm topologies in X and Y if and only if it is continuous with respect to their weak topologies.   Remark 1.2.2 It can be proved that Dunford–Schwartz theorem is also true for affine mappings. It is enough only to verify (see [38], Prop. 2.1 of Chap. I) that a mapping T : X → Y is affine if and only if there are a linear mapping L : X → Y and a point y ∈ Y such that T (x) = L(x) + y for every x ∈ X.   The most important class of linear topological spaces are locally convex spaces. A Hausdorff linear topological space X is said to be locally convex if every neighborhood of the origin includes a convex neighborhood of the origin. Let X and Y be locally convex spaces and let ·, · : X × Y → R be a functional such that (i) ·, y and x, · are continuous on X and Y for every fixed y ∈ Y and x ∈ X, respectively, (ii) if x, y = 0 for every y ∈ Y , then x = 0, and (iii) x, y = 0 for every x ∈ X, then y = 0. The pair of spaces (X,Y) with a functional ·, · is said to be a dual pair of spaces X and Y . A functional ·, · is called the duality brackets. It can be defined in X a weak topology σ (X, Y ) with respect to duality by base of neighborhoods of the origin of X of the form  U (y1 , . . . , yn ) = ni=1 {x : | x, yi | ≤ 1}, for every n ≥ 1 and y1 , . . . , yn ∈ Y . In a similar way topology σ (Y, X) in Y can be defined. It is clear that if X is a normed space, then the pair (X, X∗ ) with a functional ·, · : X × X∗ → R, defined by x, x ∗ = x ∗ (x), is a dual pair of linear spaces X and X∗ . It is easy to verify that the

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above topologies are locally convex Hausdorff topologies. Let us note that if T is a topology on X, stronger than σ (X, Y ), then every set K ⊂ X closed with respect to topology σ (X, Y ) is closed with respect to T . It can be verified that if T is a topology of locally convex space X, stronger than σ (X, Y ), then every convex set K ⊂ X closed with respect to T is closed with respect to σ (X, Y ). The following separation theorem can be proved. Theorem 1.2.2 Let X be a normed space and A, C ⊂ X be disjoint nonempty convex sets with one compact and the other closed. Then there exists a nonzero continuous linear functional x ∗ ∈ X∗ such that sup{x ∗ (a) : a ∈ A} < inf{x ∗ (c) : c ∈ C}.   Corollary 1.2.1 For every nonempty closed convex subset K of a normed space X and every non-negative numbers α1 , α2 one has (α1 + α2 ) · K = α1 · K + α2 · K. Proof It is clear that (α1 +α2 )·K ⊂ α1 ·K+α2 ·K. Suppose there is u ∈ α1 ·K+α2 ·K such that u ∈ (α1 + α2 ) · K. By Theorem 1.2.2 there exists a nontrivial linear functional x ∗ ∈ X∗ such that sup{x ∗ (x) : x ∈ (α1 +α2 )·K} < x ∗ (u), which implies that (α1 + α2 ) sup{x ∗ (x) : x ∈ K} < α1 x ∗ (x1 ) + α2 x ∗ (x2 ), where x1 , x2 ∈ K are such that u = α1 x1 + α2 x2 . Hence it follows that (α1 + α2 ) sup{x ∗ (x) : x ∈ K} < α1 sup{x ∗ (x) : x ∈ K} + α2 sup{x ∗ (x) : x ∈ K} = (α1 + α2 ) sup{x ∗ (x) : x ∈ K}, which contradicts to assumption on u. Therefore, we also have α1 · K + α2 · K ⊂ (α1 + α2 ) · K.   Remark 1.2.3 It can be proved (see [2], Theorem 8.10 of Chap. III) that if a convex set W ⊂ X with I nt (W ) = ∅ has not a common points with a convex set V ⊂ X, then there is a nontrivial functional x ∗ ∈ X∗ and a number c ∈ R such that x ∗ (x) ≤ c for x ∈ W and x ∗ (x) ≥ c for x ∈ V . If 0 ∈ I nt (W ) then we can take c = 1.   Given a set S in a normed space X can be considered as a subset of a topological space (X, σ (X, X∗ )). We will say that S is weakly closed (weakly compact) if it is closed (compact) with respect to the weak topology σ (X, X∗ ) of X. Similarly, the weak sequential closedness, the weak sequential compactness, and the weak relative sequential compactness are understood. The weak sequential closure of a set S is denoted by Seqclw (S). We have of course S ⊂ Seqclw (S) ⊂ clw (S). Given a subset S of a linear space X the convex hull of S is the intersection of all convex sets of X containing S. It is denoted by co S. If X is a linear topological space the set co S, called the closed convex hull of S, is defined to be the intersection of all closed convex subsets of X containing S. Corollary 1.2.2 Given subsets K, K1 , and K2 of a normed space (X,  · ) one has co K = cl[co K], co K = co K, and cl(co K1 + co K2 ) = cl(co K1 + co K2 ). Proof It is clear the co K is a closed subset of X such that co K ⊂ co K. Then cl[co K] ⊂ co K. But cl[co K] is a closed convex set containing K. Therefore, co K ⊂ cl[co K]. Then co K = cl[co K]. The other equalities follow from the first one and Lemma 1.2.2.   We have the following results dealing with properties of convex hulls.

1.2 Functional Analysis Preliminaries

9

Theorem (Mazur) The closed convex hull of a subset of a normed space (X,  · ) is sequentially weakly closed. Proof Let K ⊂ X and suppose a sequence (xn ) of a set W = co K is weakly convergent to x0 ∈ W . Assume x0 = 0. Since x0 ∈ W , then there is r > 0 such that a closed ball B(0, r) has not common points with W . Therefore, by Remark 1.2.3 there exists a nontrivial functional x ∗ ∈ X∗ such that |x ∗ (x)| ≤ 1 for x ∈ B(0, r) and |x ∗ (x)| ≥ 1 for x ∈ W . Hence it follows that 1 ≤ |x ∗ (xn )| → |x ∗ (x0 )| = |x ∗ (0)| = 0. Contradiction. Then for every sequence (xn ) of a set W weakly convergent to x0 we have x0 ∈ W .   Remark 1.2.4 Similarly to the proof of Corollary 1.2.2 it was verified that if K is a nonempty subset of a normed space (X,  · ), then cl[x ∗ (K)] = cl[x ∗ (K)] for every x ∗ ∈ X∗ . By Corollary 1.2.2 it follows that cl[x ∗ (co K)] = co [x ∗ (K)] for every x ∗ ∈ X∗ .   Lemma 1.2.7 For a given subset K of a linear space X, its convex hull co(K) conn sists of all convex combinations i=1 λi xi with x1 , . . . , xn ∈ K and λ1 , . . . , λn ∈ n [0, 1] such that i=1 λi = 1. n Proof Let  denotes the set of all convex combinations i=1 λi xi with n x1 , . . . , xn ∈ K and λ1 , . . . , λn ∈ [0, 1] such that λ i=1 i = 1. By n convexity of co(K) and Lemma 1.2.4 we have λ x ∈ co(K) i i i=1 n for every x1 , . . . , xn ∈ K ⊂ co(K) and λ1 , . . . , λn ∈ [0, 1] such that i=1 λi = 1. Then  ⊂ co(K). Let us observe now that  is a convex set containing K. Indeed, for every x ∈ K and λ = 1 one has x = λx + (1 − λ)z ∈  for arbitrarily taken z ∈ K. Thus  K ⊂ . For every mx, y ∈  and λ ∈ [0, 1] one has λx + (1 − λ)y = λ ni=1 λi xi + (1 − λ)  i=1 γi yi with xn1 , . . . , xn ∈ K, y1 , . . . , ym ∈ K, and λi , γi ∈ [0, 1] such that ni=1 λi = i=1 γi = 1. Let zi = xi for i = 1, . . . , n and zn+1 = y1 , . . . , zn+m = ym . Similarly, let αi = λλi for i = 1, . . . , n and αn+1 = (1 − λ)γ1 , . . . , αn+m = (1 − λ)γm . We have n+m n m α = λ λ + (1 − λ) i i i=1 i=1 γi = λ + (1 − λ) = 1. Furthermore, i=1 n+m λx + (1 − λ)y = i=1 αi zi ∈ . Then  is a convex set. Now by the definition of co(K) it follows that co(K) ⊂ . Thus co(K) = .   Corollary 1.2.3 If (X,  · ) is a normed space and (xn ) is a sequence of X weakly convergent to x0 , then some sequence of convex combinations of the elements xn converges to x0 in the norm topology of X. Proof Let (xn ) be a sequence of X weak convergent to x0 ∈ X, K = {x1 , x2 , . . .} and W = co K. A set W is a closed convex subset of X and (xn ) is its sequence weakly convergent to x0 . Therefore, x0 ∈ W , because by Mazur theorem W is sequentially weakly closed. Then there exists a sequence (zk ) of co{x1 , x2 , . . .} such that zk − x0  → 0 as k → ∞. Now by Lemma 1.2.7 the result follows.   Lemma 1.2.8 Let K and H be subsets of a linear space X. Then (a) co(α · K) = α · co(K) and co(K + H) = co(K) + co(H), If X is a linear topological space, then

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(b) co(α · K) = α · co(K), (c) co(K + H) = co(K) + co(H), if co(K) or co(H) is compact. Proof Statements (a) follows in an elementary fashion from the definition of the convex hull, Lemmas 1.2.5–1.2.7. Indeed, by Lemma 1.2.5 it follows that sets α · co(K) and co(K) + co(H) are convex sets containing α · K and K + H, respectively. Therefore, by the definition of the convex hull one has co(α · K) ⊂ α · co(K) and co(K + H) ⊂ co(K) + co(H). Now, if x ∈ co(K), by Lemma 1.2.7there are  λ1 , . . . , λn ∈[0, 1] and x1 , . . . , xn ∈ K such that ni=1 λi = 1 and x = ni=1 λi xi . Then αx = ni=1 λi (αxi ), which by Lemma 1.2.5 implies that α · coK⊂ co(α · K). Similarly, if y ∈ H and x = ni=1 λi xi ∈ co(H), then x + y = ni=1 λi (xi + y) ∈ co(K + y). But co(K) + y = co(K + y) ⊂ co(K + H) for every y ∈ H. Then co(K) + H ⊂ co(K + H). The same argument shows that co(K) + co(H) ⊂ co[co(K) + H] ⊂ co[co(K + H)] = co(K + H). This completes the proof of (a). To prove (b), let us note that co(K) is a closed convex set containing K. Thus co(K) ⊂ co(K). By virtue of Lemma 1.2.6, the closure of a convex set is convex. Then co(K) is a closed convex set containing K. Thus, co(K) ⊂ co(K), which implies that co(K) = co(K). Therefore α · co(K) = α · co(K). But α · co(K) = cl[α · co(K)], which by (a) implies that α · co(K) = co(α · K). Similarly as above we also get co(α · K) = co(α · K). Then co(α · K) = α · co(K). To prove (c), let us note that by Remark 1.2.1 and Lemma 1.2.6, a set co(K) + co(H) is closed convex and includes K + H. Therefore, co(K + H) ⊂ co(K) + co(H). Since the addition operation “+ is continuous, then co(K) + co(H) ⊂ co(K) + co(H), which by (a) and Corollary 1.2.2 implies that co(K) + co(H) ⊂ co(K + H) = co(K + H). Then co(K + H) = co(K) + co(H)   Let (X,  · ) be a Banach space. It can be proved that if S is relatively weakly sequentially compact subset of X, then clw (S) = Seqclw (S). It follows from the ˘ following Eberlein–Smulian’s theorem. ˘ Theorem (Eberlein–Smulian) Let S be a subset of a Banach space. The following statements are equivalent: (a) S is relatively weakly sequentially compact, (b) clw (S) is weakly compact, i.e., S is relatively weakly compact.

 

From the above theorem and Lemma 1.2.3 the following result follows Corollary 1.2.4 If A and B are weakly closed subsets of a Banach space, with B weakly compact, then A + B is weakly closed.   The following results deal with topological properties of convex hulls and properties of weak closures of subsets of Banach spaces. ˘ Theorem (Krein–Smulian) The closed convex hull of a weakly compact (compact) subset of a Banach space is weakly compact (compact).  

1.2 Functional Analysis Preliminaries

11

˘ Theorem (Smulian) Let S be a subset of a Banach space. If S is relatively weakly compact, then for every x ∈ clw (S) there is a sequence (xn ) of elements of S that converges weakly to x.   If X is a normed space, then X∗ is also a normed space, and hence X∗∗ (the space of all continuous linear functionals on X∗ ) is again a normed space with x ∗∗  = sup{|x ∗∗ (x ∗ )| : x ∗  = 1}. We define a linear mapping κ : X → X∗∗ , called the canonical embedding, by the equation (κx)(x ∗ ) = x ∗ (x) for all x ∗ ∈ X∗ , x ∈ X. It is clear that κ is well defined, i.e., κx = x1∗∗ and κx = x2∗∗ imply (x1∗∗ − x2∗∗ )(x ∗ ) = 0 for all x ∗ ∈ X∗ . It can be verified that the mapping κ is always one-to-one and norm preserving. Therefore, we can consider elements x of X as continuous linear functional on X∗ with x(x ∗ ) = (κx)(x ∗ ). This means that X∗ has two natural weak topologies : that generated by X∗∗ which is the weak topology for X∗ (denoted by σ (X∗ , X∗∗ )) and that generated by elements of X. The latter is called the weak ∗ - topology and denoted by σ (X∗ , X). The normed space X is said to be norm reflexive or simply reflexive if κ is onto. Remark 1.2.5 It can be proved (see [27], Th.A.3.56) that a Banach space X is reflexive if and only if X∗ is reflexive.   If X is reflexive the weak and weak ∗ - topologies for X∗ are the same, i.e., σ (X∗ , X∗∗ ) = σ (X∗ , X). If X is not reflexive the weak ∗ - topology of X∗ is weaker than its weak topology. Reflexivity is C-hereditary, i.e., every closed linear subspace of a reflexive Banach space is itself reflexive. It is easy to verify that Euclidean spaces Rd and Rd×m are both reflexive. One may easily show that a normed space, with either the weak or weak∗ - topology, is a linear topological space. Let us observe that the natural embedding κ of a reflexive space X onto X∗∗ is an isometric isomorphism between X and X∗∗ , i.e., κ(X) = X∗∗ and κx = x for every x ∈ X. We have the following important result. Theorem (Alaoglu) The closed unit ball in the conjugate space X∗ of a Banach space X is compact in its σ (X∗ , X) topology.   Corollary 1.2.5 A closed unit ball of a reflexive Banach space is weakly compact. Proof Let X be a reflexive Banach space, and κ be the natural embedding of X onto X∗∗ . The κ and κ −1 are isometrics, and κ maps the closed unit ball B of X onto the closed unit ball S of X∗∗ . It is clear from the definitions of two topologies that κ is a homeomorphism between B with its σ (X, X∗ ) topology and S with its topology σ (X∗ , X). Hence, by the Alaoglu theorem it follows that B is a weakly compact subset of the space X.   Corollary 1.2.6 A bounded subset of a reflexive Banach space is relatively weakly compact. Proof Let K be a bounded subset of a reflexive Banach space X, and let B(0, r) be a closed ball of X, centered at the origin with radius r > 0 such that K ⊂ B(0, r). It is

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clear that B(0, r) is weakly compact, because B(0, r) = r · B and by Corollary 1.2.5 the closed unit ball B of X is weakly compact. Then clw (K) is weakly compact, because it is a weakly closed subset of a weakly compact set B(0, r).   ˘ From Corollary 1.2.6, Mazur, and Eberlein–Smulian theorems the following result follows. Theorem 1.2.3 A bounded closed convex subset of a reflexive Banach space is weakly compact. Proof Let K be a bounded closed convex subset of a reflexive Banach space. By Corollary 1.2.6 a set K is relatively weakly compact. By Mazur theorem K is ˘ sequentially weakly closed, which by Eberlein–Smulian theorem implies that K is weakly closed. Thus, K is weakly compact.  

1.3 Space of Subsets of Metric Space Let (X, ρ) be a metric space and Cl(X) be the family of all nonempty closed subsets of X. Denote by Comp(X) the subfamily of Cl(X) containing all nonempty compact subsets of X. If (X, ρ) is a linear metric space, then we can also consider the family of all nonempty compact convex subsets of X. It is denoted by Conv(X). For every A, B ∈ Cl(X) we define the Hausdorff distance h(A, B) between A and B with respect to the metric ρ by setting h(A, B) = inf{ε : A ⊂ Vε (B) and B ⊂ Vε (A)}, where Vε (C) denotes the ε- neighborhood of C ∈ Cl(X), defined by Vε (C) = {x ∈ X : dist(x, C) ≤ ε} with dist(x, C) = inf{ρ(x, c) : c ∈ C}. Lemma 1.3.1 The function h : Cl(X) × Cl(X) → [0, ∞) has the following properties (a) h(A, B) = 0 if and only if A = B for A, B ∈ Cl(X), (b) h (A, B) = h(B, A) for every A, B ∈ Cl(X), (c) h(A, B) ≤ h(A, C) + h(C, B) for every A, B, C ∈ Cl(X). ¯ ¯ Proof To prove (a), let us observe that h(A, B) = max{h(A, B), h(B, A)}, where ¯h(C, D) = supx∈C dist(x, D) for C, D ∈ Cl(X). Hence it follows that h(A, B) = 0 implies that A ⊂ B and B ⊂ A, because A, B ∈ Cl(X). Then A = B. Statement (b) is evident. To prove (c) let us note that If A ⊂ Vε (C) and C ⊂ ¯ ¯ ¯ Vη (B), then A ⊂ Vε+η (B). Consequently we get h(A, B) ≤ h(A, C) + h(C, B). ¯ ¯ ¯ ¯ ¯ Thus h(A, B) = max{h(A, B), h(B, A)} ≤ max{h(A, C) + h(C, B), h(B, C) + ¯ h(C, A)} ≤ max{h(A, C) + h(C, B), h(B, C) + h(C, A)} = h(A, C) + h(C, B).   Theorem 1.3.1 (Cl(X), h) is a complete metric space if (X, ρ) is complete. Proof Let (A ) be a Cauchy sequence of Cl(X). We shall prove that the set A = ∞ ∞ n n=1 m=n Am belongs to Cl(X) and h(An , A) → 0 as n → ∞, which will be

1.3 Space of Subsets of Metric Space

13

 implied that (Cl(X), h) is a complete metric space. Let Bn = ∞ m for every m=n A n ≥ 1. Then Bn ∈ Cl(X) and Bn+1 ⊂ Bn for every n ≥ 1. Therefore, ∞ n=1 Bn ∈ Cl(X). For every ε > 0 and k ≥ 1 there exists Nkε ≥ 1 such that for every n, m ≥ Nkε one has h(An , Am ) < 2−k ε. Let (nk ) be strictly increasing sequence of N such that nk ≥ Nkε . Let x0 ∈ An0 , and select x1 ∈ An1 such that ρ(x0 , x1 ) < ε/2. In a similar way we can select points x2 , x3 , . . . , xk of sets An2 , An3 , . . . , Ank such that ρ(xi , xi+1 ) < 2−i ε for i = 1, 2, . . . , k − 1. Let us note that xk+1 ∈ Ak+1 exists, because dist(xk , Ak+1 ) ≤ h(Ank , Ak+1 ) < 2−k+1 ε. It is easy to see that (xk ) is a Cauchy sequence of the complete metric space (X, ρ). Then there is x ∈ X such that ρ(xk , x) → 0 as k → ∞. We have of course  that x ∈ A. Furthermore, for every n0 ≥ N0ε and x0 ∈ An0 we have ρ(x0 , x) ≤ ∞ i=0 ρ(xi , xi+1 )+limk→∞ ρ(xk , x) ≤ 2ε. Then for every n0 ≥ N0ε and x0 ∈ An0 one gets dist(x0 , A) ≤ ρ(x0 , x) ≤ 2ε, ¯ n0 , A) ≤ 2ε for n0 ≥ N ε . Hence it follows that h(A ¯ n0 , A) ≤ which implies that h(A 0 ε 2ε for n0 ≥ N0 . ¯ We shall verify now that h(A, An ) → 0 as n → ∞. Let ε > 0 and Nε ≥ 1 be such that h(An , Am ) ≤ ε for every n, m ≥ Nε . For every x ∈ A and n ≥ 1 one has  x∈ ∞ m=n Am . Therefore, there exists n0 ≥ Nε and y ∈ An0 such that ρ(x, y) ≤ ε. ¯ n0 , Am ) ≤ 2ε. But for each m ≥ Nε we have dist(x, Am ) ≤ dist(x, An0 ) + h(A ¯ Therefore, for every m ≥ Nε one has h(A, Am ) ≤ 2ε. Then h(An , A) → 0 as n → ∞.   Theorem 1.3.2 If (X, ρ) is a complete metric space, then Comp(X) is a closed subset of the space (Cl(X), h). Proof Let (An ) be a sequence of Comp(X) such that h(An , A) → 0 as n → ∞. Then for a given ε > 0 there exists n0 (ε) ≥ 1 such that h(An , A) < ε and A ⊂ Vε (An ) for every n ≥ n0 (ε). But An is a compact subset of the metricspace (X, ρ). Then there exists a finiteset {x1 , . . . , xn } ⊂ An such that An ⊂ ni=1 B(xi , ε), which implies that A ⊂ ni=1 B(xi , 2ε). Therefore, A ∈ Comp(X), because it is a nonempty closed bounded subset of X.   Theorem 1.3.3 If (X, ρ) is a compact metric space, then so is (Comp(X), h). Proof By [27, Th. 1.34 of Chap. 1] every sequence (An ) of Comp(X) possesses a subsequence (Ank ) convergent in the Hausdorff metric h to A ∈ Comp(X). Thus Comp(X) is a sequentially compact subset of (Cl(X), h), which is equivalent to its compactness. Then a metric space (Comp(X), h) is compact.   Corollary 1.3.1 If (X, ρ) is a compact linear metric space, then Conv(X) is a compact subset of the metric space (Comp(X), h). Proof By virtue of Theorem 1.3.3 it is enough only to verify that Conv(X) is a closed subset of (Comp(X), h). Let (An ) be a sequence of Conv(X) such that h(An , A) → n → ∞. By Theorem1.3.2 one has A ∈ Comp(X). Let  0 as ∞ ∞ ε C = ε>0 ∞ n=1 m=n Vε (Am ) and Cn = m=n Vε (Am ). It is clear that Vε (Am ) is a convex subset of X for every m > n. Therefore, Cnε is convex for every ∞ ε n ≥ 1, which implies that Cε = n=1 Cn is convex, because (Cnε ) is an increasing

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 sequence. Then C = ε>0 Cε is convex. To conclude the proof we have to verify that A = C. By limn→∞ h(An , A) = 0 it follows that for every ε > 0 there is n0 (ε) ≥ 1 such that A ⊂ Vε (Am ) and Am ⊂ Vε (A) for every m ≥ n0 (ε). Therefore,  ∞ ∞ ∞  A ⊂  ε>0 ∞ n=1 m=n Vε (Am ) = C and n=1 m=n Am ⊂ A. Finally, let ∞ ∞ x ∈ V (A ). Then for all ε > 0 there is n0 (ε) ≥ 1 such m n=1 m=n ε ε>0 that x ∈ Vε (Am ). Let n ≥ 1 be given. Then there is m ≥ max[n, n0 (ε)] such  ∞ Am ).  Hence it follows that x  ∈ ∞ that x ∈Vε (A m ) ⊂ Vε ( m=n  m=n Am and  ∞ ∞ ∞ ∞ ∞ so x ∈ ∞ n=1 m=n Am . Thus n=1 m=n Vε (Am ) ⊂ n=1 m=n Am , which together with the above inclusions implies that C ⊂ A.   Corollary 1.3.2 If (X, ·) is a normed space, then for every A, B ∈ Conv(X) one has h(A, B) = sup{|s(x ∗ , A) − s(x ∗ , B)| : x ∗  ≤ 1}, where s(·, A) and s(·, B) are support functions of A and B, respectively. Proof The result is clear if A = B. So assume that A = B and let ε > 0 be such that A ⊂ B + Bε , where Bε = {x ∈ X : x < ε}. Let us note that B + Bε = Vε (B) and A + Bε = Vε (A). Then for every x ∗ ∈ X∗ with x ∗  ≤ 1 we have s(x ∗ , A) ≤ s(x ∗ , B) + ε and s(x ∗ , B) ≤ s(x ∗ , A) + ε. Therefore, |s(x ∗ , A) − s(x ∗ , B)| ≤ ε and so sup{|s(x ∗ , A) − s(x ∗ , B)| : x ∗  ≤ 1} ≤ h(A, B). On the other hand, if ε = sup |s(x ∗ , A) − s(x ∗ , B)| > 0, then we have A ⊂ B + B ε and B ⊂ A + B ε . So h(A, B) ≤ ε, and thus the formula follows.   We can extend the definition of the Hausdorff distance on the family Pb (X) of all nonempty bounded subsets of a metric space (X, ρ). Similarly as above for every ¯ A, B ∈ Pb (X) we define h(A, B) = inf{ε > 0 : A ⊂ Vε (B)} and then the Haus¯ ¯ dorff pseudometric h on Pb (X) is defined by h(A, B) = max{h(A, B), h(B, A)} ¯ for every A, B ∈ Pb (X). It can be verified that h(A, B) = 0 if and only if A¯ = B. Theorem 1.3.4 For every A, B ∈ Pb (X) one has h(A, B) = sup{|dist(x, A) − dist(x, B)| : x ∈ X}. Proof Let us note that for every x ∈ X and b ∈ B one has dist(x, A) ≤ ρ(x, b) + dist(b, A) and therefore, dist(x, A) ≤ dist(x, B) + h(B, A). Similarly, we obtain dist(x, B) ≤ dist(x, A) + h(B, A). Then sup{|dist(x, A) − dist(x, B)| : x ∈ X} ≤ ¯ h(A, B). On the other hand, we have h(B, A) ≤ sup{|dist(b, A) − dist(b, B)| : b ∈ ¯ B} ≤ sup{|dist(x, A) − dist(x, B)| : x ∈ X} and h(A, B) ≤ sup{|dist(x, A) − dist(x, B)| : x ∈ X}. Then we also get h(A, B) ≤ sup{|dist(x, A) − dist(x, B)| : x ∈ X}.   ¯ B) ¯ ≤ h(A, B). Lemma 1.3.2 For every A, B ∈ Pb (X) one has h(A, Proof For every a ∈ A¯ and ε > 0 there is aε ∈ A such that ρ(a, aε ) ≤ ε. Therefore, ¯ ≤ ρ(a, aε ) + dist(aε , B) ¯ ≤ ε + inf{ρ(aε , b) : b ∈ B} ¯ ≤ ε + inf{ρ(aε , b) : dist(a, B) ¯ ¯ : a ∈ A} ¯ ≤ ε + h(A, ¯ b ∈ B} ≤ ε + h(A, B). Thus sup{dist(a, B) B), i.e., that ¯ A, ¯ B) ¯ ≤ ε + h(A, ¯ ¯ A, ¯ B) ¯ ≤ h(A, ¯ h( B) for every ε > 0. Then h( B). Similarly we ¯ B, ¯ A) ¯ ≤ h(B, ¯ get h( A).  

1.3 Space of Subsets of Metric Space

15

It is easy to verify that for a normed space (X,  · ) and every compact convex sets A, B ⊂ X, and λ, μ ∈ R+ one has: (i) A + {0} = {0} + A = A, (ii) (A + B) + C = A + (B + C), (iii) A + B = B + A, (iv) A + C = B + C implies A = B, (v) 1 · A = A, (vi) λ · (A + B) = λ · A + λ · B, and (vii) (λ + μ) · A = λ · A + μ · A. It can be verified that for every compact convex sets A, B, C ⊂ X such that A = B one has A + C = B + C. Let X be a normed space and A, B ∈ Pb (X). We can define the set A + (−1) · B, which is often called the Minkowski difference of sets A, B ∈ Pb (X). In the general case we have A+(−1)·A = {0}. For some nonempty compact convex sets A, B ⊂ X a difference A − B, known as the Hukuhara difference, can be defined such that A − A = {0}. More precisely, it is said that the Hukuhara difference A − B exists if there is a compact convex set C ⊂ X such that A = B + C. It is clear that if the Hukuhara difference A−B exists, then it is defined in the unique way, i.e., if there is an another compact convex set D ⊂ X such that A = B + D, then h(B + C, B + D) = 0 which by Lemma 1.3.1 implies that C = D. Lemma 1.3.3 Let (X, || · ||) be a normed space. For every A, B, C, D ∈ Pb (X) ¯ ¯ ¯ and μ ∈ R+ one has: (i) h(μA, μB) = μh(A, B), (ii) h(A + B, C + D) ≤ ¯ ¯ ¯ ¯ h(A, C) + h(B, D), and (iii) h(co A, co B) ≤ h(A, B). Proof (i) If A ⊂ Vε (B), then μA ⊂ Vμε (μB). Hence it follows that inf{η > 0 : ¯ μA ⊂ Vη (μB)} = μ inf{η > 0 : A ⊂ Vη (B)} = μh(A, B). (ii) If A ⊂ Vε (C) and B ⊂ Vη (D), then A + B ⊂ Vε+η (C + D). Therefore, inf{ε + η : A + B ⊂ ¯ ¯ Vε+η (C + D)} ≤ inf{ε : A ⊂ Vε (C)} + inf{η : B ⊂ Vη (D)} = h(A, C) + h(B, D). ¯ (iii) Let us note that for every η ≥ h(A, B) one has A ⊂ B + η · B, where B is a closed unit ball of X. It is clear that B is a convex set. Therefore, by Lemma 1.2.8 ¯ it follows that co A ⊂ co B + η · B, which implies that h(co A, co B) ≤ η, because ¯h(co A, co B) = inf{ε > 0 : co A ⊂ co B +(ε+η)·B} ≤ ε+η. Taking in particular, ¯ ¯ ¯ η = h(A, B) we get h(co A, co B) ≤ h(A, B)   In what follows, the Hausdorff sub-distance of a nonempty bounded subset A of a normed space X to the set {0} ⊂ X is denoted by A. Therefore, norms of considered normed spaces will be denoted by | · | and | · | , respectively. Then A = sup{|a| : a ∈ A} or A = sup{ | a | : a ∈ A}. The Hausdorff distance and sub-distance generated by such norms will be denoted by h, h¯ and H¯ , respectively. Corollary 1.3.3 Let (X, | · |) be a normed space and A, B ⊂ X be compact convex sets. If the Hukuhara difference A − B exists, then h(A, B) = A − B.   Let (An )∞ n=1 be a sequence of subsets of a topological space (X, T ). A point x ∈ X is called a limit point of (An )∞ n=1 if for every neighborhood U of x there is an n ≥ 1 such that for all m ≥ n one has Am ∩ U = ∅. A point x ∈ X is called a cluster point of (An )∞ n=1 if for every neighborhood U of x and every n ≥ 1 there exists an m ≥ n such that Am ∩ U = ∅. Given above the sequence (An )∞ n=1 of subsets of a topological space (X, T ) we can define its limit inferior and limit superior such as above in Section 1.1. Furthermore, we can define its topological (Kuratowski) upper and lower limits denoted by Ls An and Li An , respectively. The lower limit

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Li An is defined to be the set of all limit points of (An )∞ n=1 . It should not be confused Li An , with the limit inferior : lim inf An of (An )∞ . n=1 The upper limit Ls An is the . Similarly, Ls An should not be confused with set of all cluster points of (An )∞ n=1 the limit superior : lim sup An of (An )∞ . If Li A n = Ls An , then we say that a n=1 sequence (An )∞ has a topological (Kuratowski) limit, denoted by Lim An , equal n=1 to the common value of the limits Li An and Ls An . From the definitions of Li An ∞ and Ls An the following results follow for every sequences (An )∞ n=1 and (Bn )n=1 of subsets of a topological Hausdorff space (X, T ). Lemma 1.3.4 For every sequence (An )∞ n=1 of subsets of (X, T ) one has (a) x ∈ Li An if and only if there is an integer N ≥ 1 and a sequence (xn )∞ n=1 of X with xn ∈ An for n ≥ N such that x = limn→∞ xn , (b) x ∈ Ls An if and only if there is an increasing subsequence (nk )∞ k=1 of positive integers and a sequence (xnk )∞ of X such that x ∈ A for k = 1, 2, . . . n n k k k=1 and x = limk→∞ xnk , (c) Li An = Li An = Li An , Li An ⊂ Ls An , and lim inf An ⊂ Li An ,  (d) Ls An = Ls An = Ls An , and Ls An ⊂ ∞   n=1 An . ∞ Lemma 1.3.5 Let (An )∞ n=1 and (Bn )n=1 be sequences of subsets of (X, T ).

(a) (b) (c) (d)

If An ⊂ Bn for every n ≥ 1, then Li An ⊂ Li Bn and Ls An ⊂ Ls Bn . Ls(An ∩ Bn ) ⊂ (Ls An ) ∩ (Ls Bn ) and Li(An ∩ Bn ) ⊂ (Li A n ) ∩ (Li Bn ). If An ⊂ An+1 for n ≥ 1, then Lim An exists and Lim An = n≥1 An . If An = A for n ≥ 1, then Li An = A¯ = Ls An .

 

Corollary 1.3.4 If (X, ρ) is a metric space and An ∈ Cl(X) for every n ≥ 1, then Li An = {x ∈ X : lim[dist(x, An )] = 0}.   Theorem 1.3.5 If (X, ρ) is a compact metric space, then the sequence (An ) of Cl(X) converges in the Hausdorff metric topology to A ∈ Cl(X) if and only if Li An = A = Ls An . Proof By virtue of Lemma 1.3.1 the Hausdorff distance h on Cl(X) is a metric, because by compactness of the space (X, ρ) all elements of the space (Cl(X), h) are compact. If a sequence (An )∞ n=1 of Cl(X) converges to A ∈ Cl(X) in the Hausdorff metric, then by the definitions of the metric h and the Kuratowski limits Li An and Ls An we get A ⊂ Li An and Ls An ⊂ A. Then Li An = A = Ls An . Conversely, let A ⊂ X be such that Li An = A = Ls An . By compactness of the metric space (X, ρ) we have A = ∅. Then A ∈ Cl(X). We have to show that for every ε > 0 and sufficiently large n ≥ 1 one has An ⊂ Vε (A) and A ⊂ Vε (An ). If the first inclusion were false we would obtain a contradiction to A = Ls An . If the second inclusion were false we obtain a contradiction to Li An = A.   In what follows, we shall need necessary and sufficient conditions for the relative compactness of subsets the space C([0, T ], Conv(Rd )) of continuous functions X : [0, T ] → Conv(Rd ). They are given in the general Ascoli theorem.

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17

To present it let us recall the notion of uniform equicontinuity of subsets of the space C([0, T ], Conv(Rd )). A set H ⊂ C([0, T ], Conv(Rd )) is said to be uniformly equicontinuous if limδ↓0 sup{Vδ (X) : X ∈ H} = 0, where Vδ (X) = maxt,s∈[0,T ],|t−s|≤δ h(X(t), X(s)) for every X ∈ H. Theorem 1.3.6 (Ascoli) A set H ⊂ C([0, T ], Conv(Rd )) is relatively compact if and only if it is uniformly equicontinuous and for every t ∈ [0, T ] a set H(t) = {X(t) : X ∈ H} is a relatively compact subset of the space (Conv(Rd ), h).  

1.4 Lebesgue and Bochner Integrals Given a nonempty set T , by an σ -algebra of T we mean a family A of subsets  of T such that T ∈ A, T \ A ∈ A for every A ∈ A and n≥1 An ∈ A for every sequence (An )∞ n=1 ⊂ A. There are simpler structures of subsets of T than σ -algebras, like a ring. The family R of subsets of T is said to be a ring if for every A, B ∈ R its union A ∪ B and a difference A \ B belong to R. In particular, if R is a ring, then for every A, B ∈ R a symmetric difference A B belongs to R, because A B = (A \ A) ∪ (B \ A). In what follows, we consider σ -algebras and rings, generated by a given family E of subsets of a nonempty set T . They are denoted by σ (E) and R(E), respectively, and defined to be the smallest σ -algebra and the smallest ring, respectively, containing a family E. A set T together with its σ -algebra A is called a measurable space. It is denoted by (T , A). If (T , A) and (X, F) are measurable spaces, then a mapping f : T → X is said to be (A, F)-measurable or simply measurable if f −1 (A) ∈ A for every A ∈ F, where f −1 (A) = {t ∈ T : f (t) ∈ A}. A fundamental theorem of measure theory is known as the Monotone Class Theorem. There are a lot of versions of this theorem. We are preset here one of them. To begin with let us define first a monotone vector space. Let (T , A) be a measurable space. By a monotone vector space we mean a family H of bounded functions f : T → R such that : (i) H is a linear vector space over a real line R, (ii) constant functions belong to H and (iii) if (fn )∞ n=1 ⊂ H and 0 ≤ f1 (t) ≤ f2 (t) ≤ . . . ≤ fn (t) ≤ . . . and limn→∞ fn (t) = f (t) and f : T → R is bounded, then f ∈ H. A family M of functions f : T → R is called multiplicative if for every f, g ∈ M one has f · g ∈ M. Similarly as above for a given above family M by σ (M) we denote the σ -algebra on T generated by a family {f −1 (A) : A ∈ β(R), f ∈ M} of subsets of T . It is clear that all elements of M are σ (M)-measurable. Theorem (Monotone Class Theorem) Let M be a multiplicative class of bounded real-valued functions defined on a measure space (T , A), and let F = σ (M). If H is a monotone vector space containing M, then H contains all bounded, Fmeasurable functions.   If (T , A) is a measurable space and μ : A → R is a measure defined on A, then a system (T , A, μ) is called a measure space. It is said to be finite or

18

1 Preliminaries

σ -finite if μ is finite or σ -finite measure. In particular, if μ(T ) = 1, then a measure space (T , A, μ) is called a probability space. Then it is usually denoted by (, F, P ). Given a measurable space (T , A) and a Banach space (X, |·|) a function ν : A → X is said to be finitely additive vector measure, or simply a vector measure, if whenever E1 and of A then ν(E1 ∪E2 ) = ν(E1 )+ν(E2 ). E2 are disjointmembers ∞ If in addition ν( ∞ E ) = ν(E ) in the norm topology of X for all n n n=1 n=1 pairwise disjoint members of A, then it is termed a countable sequences (En )∞ n=1 additive vector measure. The variation of a vector measure ν : A → X is the extended non-negative function |ν| whose values on a set E ∈ A is given by  π |ν|(E) = supπ m ν(A i ), where the supremum is taken over all A-measurable i=1 partitions π = {A1 , . . . , Am π } of E, i.e., a finite numbers of pairwise disjoint π members of A such that E = m i=1 Ai . If |ν|(T ) < ∞, then ν is called a measure of bounded variation. Given a finite measure space (T , A, μ), the X-valued measure of bounded variation ν on A is said to be μ-continuous, simply denote by ν  μ, if limμ(E)→0 ν(E) = 0. A vector measure μ : A → X is said to be non-atomic if does not exist a set A ∈ A such that μ(A) = 0 and for every B ∈ A such that B ⊂ A, either μ(B) = 0 or μ(A \ B) = 0. The following property of non-atomic measures μ : A → Rd is important. Theorem (Lyapunov) Let μ : A → Rn be a non-atomic measure. Then its range R(μ) = : μ(A) is a compact convex subset of Rn .   Corollary 1.4.1 If μ : A → Rn is a non-atomic measure, then for every λ ∈ [0, 1] there exists a set Bλ ∈ A such that μ(B) = λ · μ(T ). Proof It is clear that 0 ∈ R(μ) and μ(T ) ∈ R(μ). Then by convexity of R(μ) it follows that for every λ ∈ [0, 1] one has λ · μ(T ) ∈ R(μ). Thus, there exists a set Bλ ∈ A such that λ · μ(T ) = μ(Bλ ).   In applications we have to deal with special measures called Radon measures. To define them, let us consider a topological space (T , T ) and let β(T ) be the Borel σ algebra on T . Every measure μ on β(T ) is called a Borel measure. A Borel measure μ is said to be regular, if for every set E ∈ β(T ) and every ε > 0 there exist a closed set P ⊂ T and an open subset Q ⊂ T such that P ⊂ E ⊂ Q, and such that μ(Q \ P ) < ε. The Radon measure is defined to be a set function μ : β(T ) → R such that μ = μ1 − μ2 , where μ1 and μ2 are finite regular Borel measures. In what follows, we shall assume that we have given a measure space (T , A, ν). Let (xn ) be a sequence of measurable finite a.e. functions xn : T → R. The sequence (xn ) is said to be convergent in measure to a measurable function x : T → R if lim μ({t ∈ T : |xn (t) − x(t)| ≥ ε}) = 0 for every ε > 0. It is denoted μ μ μ by xn → x. It can be verified that if xn → x and xn → y, then x = y a.e. If limn,m→∞ μ({t ∈ T : |xn (t) − xm (t)| ≥ ε}) = 0 for every ε ≥ 0, then there exists a μ measurable finite a.e. function x : T → R such that xn → x. Finally, let us note that if a sequence (xn ) of measurable functions is convergent in measure to a function x, then there is a sub-sequence of (xn ) convergent a.e. to x.

1.4 Lebesgue and Bochner Integrals

19

 Let x = (αi , Ai : i = 1, . . . , n) be a simply function, i.e., x = ni=1 1Ai αi where (Ai )ni=1 is a finite A-measurable partition of T , α1 , . . . , αn ∈ R and 1Ai is a characteristic function of a set Ai for i = 1, . . . , n. A simple function x = (αi , Ai : i = 1, . . . , n) is said to be integrable if ni=1 |αi |μ(Ai ) < ∞. For such simple function its integral with respect toa measure μ is denoted by T x(t)μ(dt) and defined by setting T x(t)μ(dt) = ni=1 αi μ(Ai ). A function x : T → R is said to be integrable or Lebesgue integrable if there is a sequence (xn ) of simple functions convergent a.e. to x and such that  limn,m→∞ T |xn (t) − xm (t)|μ(dt = 0. It can be verified that if x : T → R is  integrable, then a limit lim T xn (t)μ(dt) does not depend on a sequence (xn ) of  simple functions satisfying conditions mentioned above. The limit lim x (t)μ(dt n T  is denoted by T x(t)μ(dt) and said to be a Lebesgue integral of x with respect to a measure μ. In such case x is said to be μ-integrable. It can be verified that every integrable function is measurable and finite a.e. If a measurable function x : T → R andA ∈ A are such that a function 1A x is μ-integrable, then its integral is denoted by A x(t)μ(dt) and said to be the Lebesgue integral of x over the set A. Given a μ-integrable function f : T → R we can define on A the mapping  ϕ(f ) : A  A → A f (t)μ(dt) ∈ R. A mapping ϕ is absolutely continuous with respect to μ, i.e., ϕ(An ) → 0 for every sequence (An ) ⊂ A such that μ(An ) → 0. Conversely, we have also the following theorem. Theorem (Radon–Nikodym) If μ is σ -finite measure on A and ν is a measure absolutely continuous with respect to μ, then there exists the unique with respect to equality a.e. measurable and non-negative  a.e. function f : T → R, called the density of a measure ν, such that ν(A) = A f (t)μ(dt) for every A ∈ A.   The Radon–Nikodem theorem can be also proved (see [94]) for mappings with values in Banach spaces. Apart from the mapping ϕ defined above, we can define a mapping that associate to each μ-integrable function f : T → R  its integral T f (t)μ(dt). It can be verified that such defined mapping is linear on an appropriate defined linear space of μ-integrable functions. In particular, if f : T → R is a positive a.e. μ-integrable function, then ϕ(f ) : A → R+ is a measure on A. It is non-atomic if μ is a non-atomic measure. In what follows, the following result will be needed. Lemma 1.4.1 Let (T , A, μ) be a non-atomic measure space and f : T → R  be μ-square integrable, i.e., such that T f 2 (t)μ(dt) < ∞. Then there exists an orthogonal sequence (fn )∞ of μ-square integrable functions fn : T → R such   n=1 that T fn2 (t)μ(dt) = T f 2 (t)μ(dt) and fn (t) ∈ {−f (t), f (t)} for n = 0, 1, 2, . . . and μ-a.e. t ∈ T . Proof In order to define the required sequence we shall construct a sequence 0 ({C1n , C2n , . . . , C2nn })∞ n=0 of finite A-measurable partitions of T such that C1 = : T and C10 = C11 ∪ C21 , C11 = C12 ∪ C22 , C21 = C32 ∪ C42 ,

20

1 Preliminaries

C12 = C13 ∪ C23 , C22 = C33 ∪ C43 , C32 = C53 ∪ C63 , C42 = C73 ∪ C83 , ............................................... n+1 C1n = C1n+1 ∪ C2n+1 , C2n = C3n+1 ∪ C4n+1 , . . . , C2nn = C2n+1 n+1 −1 ∪ C2n+1

and such that 

 f (t)μ(dt) = (1/2 ) 2

Ckn

f 2 (t)μ(dt).

n

T

for every k = 1, 2, . . . , 2n and n = 0, 1, 2, . . .. μ0,1 be a measure on A defined for every B ∈ A by setting μ0,1 (B) =  Let 2 (t)μ(dt). It is clear that μ f 0,1 is non-atomic and therefore, by Corollary 1.4.1 B 1 there exists C1 ∈ A such that μ0,1 (C11 ) = (1/2)μ0,1 (C10 ). Taking now, C21 = C10 \ C11 we get μ0,1 (C10 ) = μ0,1 (C11 ) + μ0,1 (C21 ), which implies that μ0,1 (C11 ) =  μ0,1 (C21 ) = (1/2) T f 2 (t)μ(dt). Suppose now the above procedure is satisfied for family C1n , C2n , . . . , C2nn . Applying it again to Ckn , instead of to C10 , with a non atomic measure μn,k defined by μn,k (B) = C n ∩B f 2 (t)μ(dt) for B ∈ A and k k = 1, 2, . . . , 2n , we get the above relations between sets C1n , C2n , . . . , C2nn and elements of the family C1n+1 , C2n+1 , . . . , C2n+1 n+1 . Then by the inductive procedure the above relations are satisfied for every n ≥ 1. Let us define now the required sequence (fn )∞ n=1 by setting n

fn =

2 

(−1)k−1 1Ckn f for n = 0, 1, 2, . . . .

k=1

n It is clear that fn2 = 2k=1 1Ckn f 2 for n=0, 1, 2, . . ., because ({C1n , C2n , . . . , C2nn })∞ n=0  is a sequence of A-measurable partitions of T . Then C n f 2 (t)μ(dt) = k  (1/2n ) T f 2 (t)μ(dt). Finally, from the definition of functions fn , one has fn (t) ∈ {−f (t), f (t)} for n = 0, 1, 2, . . . and μ-a.e. t ∈ T , because fn (t) = [1C1n f (t) − 1C2n f (t)] + . . . + [1C nn f (t) − 1C nn f (t)] for n = 0, 1, 2, . . . 2 −1 2 and μ-a.e. t ∈ T .   Remark 1.4.1 For example, if T = [0, 1], f ≡ 1, then the functions constructed in Lemma 1.4.2 may be taken equal a.e. to the Rademacher functions rn (t) = sin(2n π t) for n ≥ 0. For instance, fn (t) = rn (t)1[0,1]\{0} (sin(2n π t)) +   1{0} (sin(2n π t)) for a.e. t ∈ [0, 1]. The following results are applied in the sequel. Theorem (Monotone Convergence Theorem) If (xn ) is a non-increasing sequence of integrable functions   xn : T → R convergent a.e. to a function x, then limn→∞ T xn (t)μ(dt) = T x(t)μ(dt).  

1.4 Lebesgue and Bochner Integrals

21

Theorem (Lebesgue) If (xn ) is a sequence of integrable functions convergent a.e. + to a function x, and if there exists an integrable function m : T  → R such that|xn (t)| ≤ m(t) for a.e. t ∈ T , then x is μ-integrable and T x(t)μ(dt) =   lim T xn (t)μ(dt). Lemma (Fatou) If (xn ) is a sequence of non-negative measurable functions  convergent a.e. to a function x, then T x(t)μ(dt) ≤ lim inf T xn (t)μ(dt).   Lemma (Gronwall) For every positive numbers α, β ∈ R and positive Lebesgue t integrable functions u, v : [0, ∞) → R such that u(t) ≤ α + β 0 u(τ )v(τ )dτ for t every 0 ≤ t < ∞, one has u(t) ≤ α exp{β 0 v(τ )dτ } for every 0 ≤ t < ∞. t Proof From the inequality u(t) ≤ α + β 0 u(τ )v(τ )dτ it follows ln(α + t t β 0 u(τ )v(τ )dτ ) − ln(α) ≤ β 0 v(τ )dτ for every 0 ≤ t < ∞. Then α + t t β 0 u(τ )v(τ )dτ ≤ α exp{β 0 v(τ )dτ , which by the inequality u(t) ≤ α + t t   β 0 u(τ )v(τ )dτ implies u(t) ≤ α exp{β 0 v(τ )dτ } for every 0 ≤ t < ∞. We can extend the definition of Lebesgue integrals to the case of vector-valued functions x : T → Rr . If x = (x1 , . . . , xr ) and all real-valued functions xi are μ integrable for i = 1, . . . , r, then x is said to be μ-integrable, and x(t)μ(dt) = T   ( T x1 (t)μ(dt), . . . , T xr (t)μ(dt)). It can be verified that for μ-integrable vector  valued functions x, y, z and α, β ∈ R one has | T z(t)μ(dt)| ≤ T |z(t)|μ(dt) and  T (αx(t) + βy(t)) μ(dt) = α T x(t)μ(dt) + β T y(t)μ(dt). Remark 1.4.2 It is clear that if μ1 and μ2 are measures on a σ -algebra A of subsets of T , then μ1 +μ2 : A → R+ defined by (μ1 +μ2 )(A) = μ1 (A)+μ2 (A) for every A ∈ A, is a measure on A. If f : T → Rn is μ1 - and μ2 - Lebesgue integrable,  then it is also (μ + μ ) integrable and f (t)(μ + μ )(dt) = f (t)μ 1 2 1 2 1 (dt) + T T  T f (t)μ2 (dt). The integration can be also defined for vector-valued functions defined on a measure space (T , A, μ) with values in a Banach space (X, | · |). There are a lot of possibilities for definitions of integrals for such functions. The most important is the Bochner integral, defined for strong measurable vector-valued functions x : T → X  such that T |x(t)|μ(dt) < ∞. A vector-valued function x : T → X is said to be strong measurable or μ-measurable, if there exists a sequence (xn ) of simply functions xn : T → X such that xn (t) → x(t) for μ-almost all t ∈ T . Similarly to the theory of Lebesgue integral,  a vector function x : T → X is called a μsimply function if it is equal to ni=1 1Ei xi , μ-a.e., where x1 , . . . , xn ∈ X and {E1 , . . . , En } is an F-measurable partition of T . Similarly as above, it is denoted by x = (xi , Ei : i = 1, . . . , n). A vector-valued function x : T → X is said to  be Bochner integrable if it is strong measurable and T |x(t)|μ(dt) < ∞. To define the Bochner integral for Bocher integrable vector-valued functions we begin with the following lemmas.

22

1 Preliminaries

Lemma 1.4.2 If x : T → X is Bochner integrable, then there is a sequence (xn ) of simply functions xn : T → X such that |xn (t)| ≤ 2|x(t)| and |xn (t) − x(t)| → 0 for a.e. t ∈ T . Proof There exists a sequence (yn ) of simple functions such that |yn (t)−x(t)| → 0 a.e. Therefore, |yn (t)| → |x(t)| a.e. Let A = {t ∈ T : x(t) = 0}, Bn = {t ∈ T : |yn (t)| ≤ 2|x(t)|}, and An = {t ∈ T : |yn (t)| > 2|x(t)|}. It is clear that A, Bn , and An are measurable such that T = A ∪ Bn ∪ An and T \ A = ∞ n=1 (Bn \ A). Let  xn (t) =

yn (t) for t ∈ Bn \ A 0 for t ∈ A ∪ An .

The functions xn are simple and |xn (t)| ≤ 2|x(t)| a.e. For t ∈ A we have xn (t) = x(t), and therefore |xn (t) − x(t)| → 0 for t ∈ A. If t ∈ T \ A, then |x(t)| = 0. Therefore, |xn (t)| < 2|x(t)| which implies that there is m ≥ 1 such that |yn (t)| < 2|x(t)| for n ≥ m. Therefore, for t ∈ T \ A one has t ∈ ∞ n=1 (Bn \ A) and yn (t) = xn (t), which implies that |xn (t) − x(t)| → 0 as n → ∞.    If x = (xi , Bi : i = 1, . . . , n) is the simple function such that |x(t)| ≤ m(t) and T m(t)μ(dt) < ∞, then  |x(t)|μ(dt) = T

n  i=1

 |xi |μ(Bi ) ≤

m(t)μ(dt). T

Hence it follows that if for any i ∈ {1, . . . , n} we have μ(Bi ) = ∞, then for such i it has to be xi = 0. If we assume that 0 · ∞ = 0, then for the abovesimple function x we can define its Bochner integral (B) T x(t)μ(dt) equals to ni=1 xi μ(Bi ). It is clear that such defined integral is a linear operation on the space   of all simple Bochner integrable functions. Furthermore, |(B) T x(t)μ(dt)| ≤ T |x(t)|μ(dt). Lemma 1.4.3 If (xn ) is a sequence of simple functions xn : T → X convergent a.e. to zero and such that |xn (t)| ≤ m(t) a.e. with T m(t)μ(dt) < ∞, then  lim (B) T xn (t)μ(dt) = 0.   We can define now the Bochner integral for the Bochner integrable vectorvalued function x : T → X. If (xn ) is a sequence of simple functions  convergent a.e. to x and |xn (t)| ≤ m(t) a.e. with T m(t)μ(dt)  < ∞, then the Bochner integral (B) x(t)μ(dt) of x is defined by (B) T T x(t)μ(dt) =  lim{(B) x (t)μ(dt)}. It is clear that the above limit exists, because n  T   |(B) T xm (t)μ(dt) − (B) T xn (t)μ(dt)| ≤ T |xm (t) − xn (t)|μ(dt), |xm (t) − xn (t)| ≤ 2m(t) a.e. and |xm (t) − xn (t)| → 0 a.e. if n, m → ∞. From this definition it follows that the above-defined integral possesses the same properties as the Bochner integrals of simple functions. Given a measurable set A ⊂ T a vector-valued function x : T → X is Bochner integrable over a set A if the function 1A x is Bochner integrable. If it is the case, then a Bochner integral

1.4 Lebesgue and Bochner Integrals

23

 of x over a set A is denoted by (B) A x(t)μ(dt). It can be easily verified that if {A1 , . . . An } is a family of disjoint subsets of measurable subsets of T such that x : T → X is integrable over Ai for  Bochner  every i = 1, . . . , n, then (B) A x(t)μ(dt) = ni=1 Ai x(t)μ(dt), where A = ni=1 Ai . We shall need the following property of Bochner integrals. Theorem 1.4.1 Let X and Y be Banach spaces and let L : X → Y be a linear continuous mapping. If x : T → X is a Bochner integrable function, then L ◦ x :  T → Y is Bochner integrable and (B) T L[x(t)]μ(dt) = L[(B) T x(t)μ(dt)].   The following convergence theorems are true for Bochner integrals. Theorem (Vitali) Let (T , A, μ) be a finite measure space and (xn ) a sequence of Bochner integrable vector-valued functions xn : T → X convergent a.e. to a  function x : T → X. If furthermore |x (t)|μ(dt) → 0 as μ(E) → 0, then n E  lim E |xn (t) − x(t)|μ(dt) = 0 for every E ∈ A.   Theorem (Lebegue) Let (xn ) be a sequence of Bochnerintegrable functions xn : T → X convergent a.e. to x and |xn (t)| ≤ m(t) a.e. with T m(t)μ(dt) < ∞. Then (B) T x(t)μ(dt) = lim (B) T xn (t)μ(dt).   There is the infinite dimensional version of Lyapunov theorem. It has been proved by J.J. Uhl (see [13, 94]) and it is much complicated, then the classical Lyapunov theorem. It is connected with Banach spaces having the Radon–Nikodym property. Let us recall that the Banach space X is said to have the Radon–Nikodym property, denote simply by RNP, if for each finite measure space (T , A, μ) and each X-valued measure ν on A which is of bounded variation and μ-continuous,  there exists a Bochner integrable function f : T → X such that ν(A) = (B) A f (t)μ(dt) for every A ∈ A. It is known (see [27], Th.A.3.94, Th.A.95, p.917) that the following classes of Banach spaces have the RNP: (i) reflexive spaces and (ii) separable dual spaces. Remark 1.4.3 It can be proved (see [27], Prop.A.3.97) that for a separable Banach space X, X∗ has RNP if and only if it is separable.   Theorem (Uhl) If (X, | · |) is a Banach space having the RNP and μ : A → X is a non-atomic vector measure of bounded variation, then cl[R(μ)] is a compact convex subset of X, where R(μ) = : μ(A).   Finally, we shall present some remarks dealing with the conjugate space to the normed space C(S, X) of continuous function defined on a compact metric space (S, ρ) with values in a normed space (X, | · |) and the supremum norm | · | . It is a Banach space if X is a Banach space. Then for every linear continuous functional x ∗ on C(S, X) there is exactly  one Radon measure μ on the Borel σ -algebra β(S) of S and such that x ∗ (x) = S x(s)μ(ds)  for every x ∈ C(S, X), and conversely, each functional x ∗ defined by x ∗ (x) = S x(s)μ(ds) for every x ∈ C(S, X) and every Radon measure μ on the Borel σ -algebra β(S) is linear and continuous. Then the conjugate space C ∗ (S, X) to C(S, X) can be treated as the space of all

24

1 Preliminaries

Radon measures on the Borel σ -algebra β(S), which implies that a sequence (xn ) of C(S, X) converges weakly to x ∈ C(S, X) if and only if supn≥1 | xn | < ∞ and limn→∞ |(xn (s) − x(s)| = 0 for every s ∈ S.

1.5 Random Variables Let (, F, P ) be a probability space. In economical applications it is considered together with increasing family F = (Ft )t≥0 of sub-σ -algebras Ft ⊂ F, called a filtration. In such a case the probability space (, F, P ) with its filtration F is denoted by (, F, F, P ) and said to be a filtered probability space. It is called complete if P is a complete measure, i.e., if 2B ⊂ F for every B ∈ F such that P (B) = 0 . We say that a filtration Fsatisfies the usual conditions if F0 contains all P -null sets of F and Ft = ε>0 Ft+ε for every t ≥ 0 . If the last condition is satisfied we say that a filtration F is right continuous. We call a filtration F left continuous if Ft is generated by a family {Fs : 0 ≤ s < t} for every t ≥ 0, i.e., Ft = σ ({Fs : 0 ≤ s < t}) for every t ≥ 0. A filtration F is said to be continuous if it is right and left continuous. From the practical point of view a filtered probability space PF = (, F, F, P ) is usually regarded as a probability model of a given experiment with results belonging to . The family F is then referred to all information on elements of , whereas the filtration contains all information on elements of  given up to time t ≥ 0. The following results are important in applications. ∞ Lemma (Borel–Cantelli) (, F, P ) be a probability Let ∞ and (An )n=1 be a ∞ space ∞ sequence of F such that n=1 P (An ) < ∞. Then P ( m=1 n=m Am ) = 0.  

Theorem (Halmos) If μ is σ -finite measure on a ring R, then for every set E of finite measure in σ (R) and every positive number ε there exists a set Eε in R such   that μ(E Eε ) ≤ ε. We can prove now the following result. Lemma 1.5.1 Let T > 0, 0 ≤ s < t ≤ T , n ≥ 1 and δn = T /2n be such that s < t − δn for n ≥ n. Assume PF = (, F, F, P ) is a filtered probability space with a continuous filtration F = (Ft )t≥0 . Then for every ε > 0 and every partition nε ≥ n such that for every n ≥ nε there exists a (Ak )N k=1 ∈ (, Ft ) there is an +1 ∈ (, F ) such that P (B ) ≤ ε and P (A partition (Bk )N t−δn N +1 k Bk ) < ε/N k=1 for every k = 1, 2, . . . , N, where (, Ft ) and (, Ft−δn ) denote all finite Ft -, and Ft−δn -measurable, respectively, partition of . Proof Let T > 0, δn = T /n, and 0 ≤ s < t ≤ T be fixed. It is clear that there exists n ≥ 1 such that s < t − δn for n ≥ n, because δn → 0 as n → ∞. Let (Ak )N ∈ (, F ) be a given partition. By continuity of the filtration t k=1  ∞ F, one has Ft = σ ( ∞ n=1 Ft−δn ). It is clear that R = n=1 Ft−δn is a ring of subsets of . Then by Halmos theorem for every Ak ∈ Ft and ε > 0 there exists

1.5 Random Variables

25

 2 k ∈ ∞ B n=1 Ft−δn such that P (Ak Bk ) < ε/(3N − 2N ). Therefore, for every ε k ∈ Ft−nε . Let k = 1, 2, . . . , N there exists nk ≥ 1 such that B nε = max{ n, nεk : k k ∈ Ft−δn for every k = 1, 2, . . . , N}. For every n ≥ nε one has s < t − δn and B 2 k = 1, 2, . . . , N . In particular, we have P (A1 B1 ) < ε/(3N − 2N ) < ε/2N . 1 and Bk = B k \ k−1 B Let B1 = B j =1 j for every k = 2, 3, . . . , N. It is clear that N N  k \ [( k−1 B k and Bk = B Bi ∩ Bj = ∅ for i = j , k=1 Bk = k=1 B j =1 j ) ∩ Bk ]  N for k = 2, 3, . . . , N and i, j ∈ {1, 2, . . . , N}. Let BN +1 =  \ k=1 Bk . Then N +1 k=1 Bk =  and Bi ∩ Bj = ∅ for i = j with i, j ∈ {1, 2, . . . , N + 1}. Thus, +1 nε . By the definition of B1 and properties of (Bk )N k=1 ∈ (, Ft−δn ) for every n ≥ 1 ) < ε/2N . To obtain the estimation P (Ak Bk ) < ε/N 1 it follows that P (A1 B B i ∩ B k ) < 3ε/(3N 2 − 2N ) for for every k = 2, 3, . . . , N let us note first that P (B every i = k with i, k ∈ {2, 3, . . . , N }. Indeed, let us note that for every i = k with i, k ∈ {2, 3, . . . , N } one has

N  P (Bi ∩ Bk ) = P [Bi ∩ Bk ] ∩ Aj =

(1.5.1)

j =1

i ∩ B k ] ∩ P (Bi ∩ Bk ∩ Ai ) + P (Bi ∩ Bk ∩ Ak ) + P [B

N 

Aj .

j =i,j =k

i \ Ai ) < ε/(3N 2 − 2N ) and P (Ai \ But, for every i = 2, 3, . . . , N one has P (B 2 Bi ) < ε/(3N − 2N), because P (Ai Bi ) < ε/(3N 2 − 2N ) for i = 1, 2, . . . , N . Therefore,

i ∩ B k ] ∩ P [B

N  j =i,j =k

Aj



i ∩ ≤P B

N 

Aj

=

(1.5.2)

j =i

2 i ∩ A∼ P (B i ) = P (Bi \ Ai ) < ε/(3N − 2N ).

Furthermore, for k = i we have Ai ⊂ A∼ k . Then for every k = i one has Bk ∩ Ai ⊂ k \ Ak . Therefore, P (B i ∩ B k ∩ Ai ) ≤ P (B k ∩ Ai ) ≤ P (B k \ Ak ) < k ∩ A∼ = B B k i ∩ B k ∩Ak ) ≤ ε/(3N 2 −2N ) for every k = i. Similarly, for every k = i one gets P (B 2 P (Bi ∩ Ak ) < ε/(3N − 2N). Hence, by (1.5.1) and (1.5.2), for every k = i we i ∩ B k ) < 3ε/(3N 2 − 2N). obtain P (B Let us note now that for every k = 2, 3, . . . , N we have 

k−1 k−1    k ) ∪ Ak ∩ B k ∩ j j . k \ B k ∩ = (Ak \ B Ak \ B B B j =1

j =1

26

1 Preliminaries

Hence, and the definition of Bk , for every k = 2, 3, . . . , N we get P (Bk Ak ) = P (Bk \ Ak ) + P (Ak \ Bk ) =



k−1 k−1     j j k \ B k ∩ k ∩ + P Ak \ B ≤ P B B B j =1

j =1

k−1  k \ Ak ) + P (Ak \ B k ) + P Ak ∩ B k ∩ j ≤ ε/(6N 2 − 4N )+ P (B B j =1



k ∩ P B

k−1 

j B

j =1

≤ ε/(6N − 4N) + 2

k−1 

k ∩ B j ) ≤ P (B

j =1

ε/[(6N − 4N)] + 3(k − 1)ε/(3N 2 − 2N ) ≤ 

3(N − 1) 3N − 2 1 + = ε/N. =ε ε 2N (3N − 2) N(3N − 2) N(3N − 2) 2

It was noted above, that we also have P (B1 A1 ) ≤ ε/N. Thus, for every k = 1, 2, . . . , N one has P (B  k Ak ) ≤ ε/N. Finally, N by the definition N of BN +1 we get P (BN +1 ) = P ( \ N B ) = E|1 − 1 | = E| k  B k=1 k=1 k k=1 1Bk − N N N N 1 | ≤ E|1 − 1 | = P (B

A ) ≤ ε/N = N ε/N = Bk Ak k k k=1 Bk k=1 k=1 k=1 ε.   Given a probability space (, F, P ) and a metric space (S, ρ) by an S-random variable on PF we mean an (F, β(S))-measurable mapping X :  → S, i.e., such that X−1 (A) ∈ F for every A ∈ β(S), where as usual β(S) denotes the Borel σ -algebra on S and X−1 (A) = {ω ∈  : X(ω) ∈ A} . We shall also say that X is a random variable on (, F, P ) with values at S. In particular if S = Rn or S = Cl(Rn ), then S-random variables are also called an n-dimensional or set-valued random variables, respectively. If S is different from Rn , then S-random variables are also called S-random elements. The σ -algebra generated by X is denoted by FX and defined to be the smallest σ -algebra on  containing all sets X−1 (U ) for all open sets U ⊂ S. It is easy to see that FX = {X−1 (A) : A ∈ β(S)}. For a given n-dimensional random variable X and a σ -algebra G ⊂ F, by EX and E[X|G] we denote mean value and a conditional expectation of X with respect to G, respectively. Let us recall that for a given P -integrable random variable X : n   → R and a σ -algebra G ⊂ F a mean value EX is defined by setting EX = E[X|G] is a G-measurable random variable  XdP , and a conditional expectation  such that H E[X|G]dP = H XdP for every H ∈ G. The following Chebyshev’s inequality P ({|ξ |γ > a}) ≤ 1/a γ E|ξ |γ is satisfied for every a ≥ 0, 0 < γ < ∞ and a random variable ξ :  → Rd such that E|ξ |γ < ∞.  from the inclusion {ω ∈  : |ξ | > γ } ⊂  and the  Indeed, it follows inequality {ω∈:|ξ |>γ } dP ≤  |ξ |γ dP .

1.5 Random Variables

27

Every random variable X :  → S induces a probability measure μX on the Borel σ -algebra β(S) of a metric space (S, ρ). It is denoted by P X−1 and called the distribution of X. It is defined by μX (A) = P (X−1 (A)) for A ∈ β(S). Lemma 1.5.2 Let (S, ρ) and (Y, G) be a metric and a measurable space, respectively, and let  : S → Y be an (β(S), G)-measurable mapping. If X and , P are S-random variables defined on probability spaces (, F, P ) and  ), , F X −1 , then P ( ◦ X)−1 = P ( ◦ X) −1 . respectively, and such that P X−1 = P X = ◦X . For every A ∈ G one has P ({Z ∈ A}) = Proof Let Z =  ◦ X and Z −1 −1 (X −1 (−1 (A))) = P ({ ◦ X ∈ A}) = P ({ ◦ X ∈ A}) = P (X ( (A))) = P −1 −1 P ({Z ∈ A}) . Then P ( ◦ X) = P ( ◦ X) .   P ) be probability spaces, X :  → , F, Corollary 1.5.1 Let (, F, P ) and ( : → C([0, T ], S) be random elements such that P X−1 = C([0, T ], S) and X −1 . Then P Xt−1 = P X t−1 for every t ∈ [0, T ], and X t = Z , P -a.s. for every PX (t, Z) ∈ [0, T ] × S such that Xt = Z , P -a.s. Proof The result follows from Lemma 1.5.2. Indeed, let t ∈ [0, T ] be fixed and et : C([0, T ], S)  X → Xt ∈ S. It is clear that et is continuous, et ◦ X = Xt and =X t . From Lemma 1.5.2 it follows P Xt−1 = P (et ◦ X)−1 = P (et ◦ X) −1 = et ◦ X −1 P Xt . Let (t, Z) ∈ [0, T ] × S be fixed and Xt = Z , P -a.s. By the equality −1 it follows that P (et ◦ X)−1 (A) = P (et ◦ X) −1 (A) P (et ◦ X)−1 = P (et ◦ X) for A = {Z} ⊂ S. Then P ({Xt = Z}) = P ({Xt = Z}), which implies that ({X t = Z}) = 1, because P ({Xt = Z}) = 1. P   From Radon–Nikodym theorem, it follows that distributions of n-dimensional random variables can be characterized by their densities. The most important are real-valued random variables X :  → R with densities of the form pX (x) = √ 1/σ 2π · exp(−(x − m)2 /2σ 2 ), where σ > 0 and m are constant, known as Gaussian or normal random variables. In particular, if m = 0 and σ = 1, then X is called a standard Gaussian variable. This notation can be also applied to n-dimensional random variables X :  → Rn . In such case the density pX is defined by √

|C| 1  pX (x1 , . . . , xn ) = · exp − · (xj − mj )aj k (xk − m)k) , 2 (2π )n/2 j,k

where m = (m1 , . . . , mn ) ∈ Rn and C = (aj k )n×n is a symmetric positive definite matrix. The following results (see [74]) will be used in the sequel. Theorem (Sudakov–Slepian–Fernique) Let {X1 , . . . , XN } and {Y1 , . . . , YN } be families of real-valued centered Gaussian random variables. If E(|Xi − Xj |2 ) ≥ E(|Yi − Yj |2 ) for i, j = 1, 2, . . . , N, then E(max{X1 , . . . , XN }) ≥ E(max{Y1 , . . . , YN }.

28

1 Preliminaries

Theorem (Fernique) Let X1 , X2 , . . . be a sequence of identically distributed standard Gaussian variables. Then there exists a universal constant K > 0 such that for all N ≥ 1 one has E(max{X1 , . . . , XN }) ≥ K[ln(N )]1/2 .   Let (S, ρ) be a separable metric space and β(S) the Borel σ -algebra on S. Denote by M(S) the space of all probability measures on β(S) and let Cb (S) be the space of all continuous bounded functions f : S → R. We say that a sequence ∞ (Pn )n=1 of M(S) weakly converges to P ∈ M(S) if limn→∞ S f dPn = S f dP for every f ∈ Cb (S). We shall denote this convergence by Pn ⇒ P . Theorem 1.5.1 The following conditions are equivalent to the weak convergence of a sequence (Pn )∞ n=1 of M(S) to P ∈ M(S) : (a) lim supn→∞ Pn (F ) ≤ P (F ) for every closed set F ⊂ S, (b) lim infn→∞ Pn (G) ≥ P (G) for every open set G ⊂ S.

 Proof Let  Pn ⇒ P . Hence it follows that lim supn→∞ Pn (F ) ≤ limn→∞ S fk dPn = S fk dP for every closed set F ⊂ S , where fk (x) = ψ(k · dist(x, F )) with ψ(t) = 1 for t ≤ 0 , ψ(t) = 0 for t ≥ 1 and ψ(t) = 1 − t for 0 ≤ t ≤ 1 . Passing in the above inequality to the limit with k → ∞ we see that (a) is satisfied. It is easy to see that (a) is equivalent to (b). Indeed, by virtue of (a) for every open set G ⊂ S we obtain lim supn→∞ Pn (S \ G) ≤ P (S \ G) , which implies that lim infn→∞ Pn (G) ≥ P (G) . In a similar way we can see that from (b) it follows that lim supn→∞ Pn (F ) ≤ P (F ) for every closed set F ⊂ S. Assume (a) is satisfied and let f ∈ Cb (S). We can assume that 0 < f (x) < 1 for x ∈ S. Then k  i−1 i=1

k

   i−1 i ·P x ∈S : ≤ f (x) < ≤ f (x)dP ≤ k k S   k  i−1 i i ·P x ∈S : ≤ f (x) < . k k k i=1

For every F side of the above inequality i = {x ∈ S : i/k ≤ f (x)} the right-hand k−1 is equal to k−1 P (F )/k and the left-hand side to i=0 n i i=0 Pn (Fi )/k − 1/k . This and (a) imply  f (x)dPn ≤ lim sup

lim sup n→∞

S

n→∞

k−1  i=0

Pn (Fi )/k ≤

k−1  i=0

 P (Fi )/k ≤ 1/k +

f (x)dP . S

  with Then lim supn→∞ S f (x)dPn ≤ S f (x)dP . Applying  the above procedure  the function g = 1 − f we obtain lim infn→∞ S f (x)dPn ≥ S f (x)dP . Therefore,

1.5 Random Variables

29



 f (x)dP ≤ lim inf

S

Thus limn→∞

n→∞

 S



 f (x)dPn ≤ lim sup

S

f (x)dPn =

 S

n→∞

f (x)dPn ≤ S

f (x)dP . S

f (x)dP for every f ∈ Cb (S).

 

We can consider weakly compact subsets of the space M(S). Let us note that on M(S) can be defined a metric d such that weak convergence in M(S) of a sequence (Pn )∞ n=1 to P is equivalent to d(Pn , P ) → 0 as n → ∞. Therefore, we say that a set  ⊂ M(S) is relatively weakly compact if every sequence (Pn )∞ n=1 of  possesses a subsequence (Pnk )∞ k=1 weakly convergent to P ∈ M(S). If P ∈ , then  is called weakly compact. We shall prove that for relative weak compactness of a set  ⊂ M(S) it suffices that  be tight, i.e., that for every ε > 0 there exist a compact set K ⊂ S such that P (K) ≥ 1 − ε for every P ∈ . Theorem 1.5.2 Every tight set  ⊂ M(S) is relatively weakly compact. Proof Assume first that (S, ρ) is a compact metric space. By the Riesz theorem we have M(S) = {μ ∈ C ∗ (S) : μ(f ) ≥ 0 for f ≥ 0 and μ(1) = 1}, where 1(x) = 1 for x ∈ S and C ∗ (S) is a dual space of C(S). Since C(S) = Cb (S), then weak convergence of probability measures is in this case equivalent to convergence on C ∗ (S) with respect to weak ∗ -topology. Then M(S) is weakly compact, because every weakly ∗ - closed subset of the unit ball of C ∗ (S) is weakly ∗ - compact. In the general case let us note that S is homeomorphic to a subset of a compact metric space. In fact it is a subset of the cube [0, 1]N . Therefore, we can assume that S is a subset of a compact metric space S. For every probability measure μ = on (S, β(S)) let us define a probability measure μ on ( S, β( S)) by setting μ(A) μ(A ∩ S) for A ∈ β(S). Let us observe that A ⊂ S belongs to β(S) if and only if ∩ S for any A ∈ β( A=A S). We shall now show that if  ⊂ M(S) is tight, then every sequence (μn )∞ n=1 of  possesses a subsequence weakly convergent to μ ∈ M(S) . Assume that a sequence (μn )∞ μn )∞ n=1 is given and let ( n=1 be sequence of probability measures ∩ S) for A ∈ β( S) and n ≥ 1. It is clear defined on β(S) by taking μn (A) = μn (A ∞ that a sequence ( μn )n=1 possesses a subsequence ( μnk )∞ k=1 weakly convergent to a probability measure ν on ( S, β( S)). We shall show that there exists a probability measure μ on (S, β(S)) such that μ = ν and that a subsequence (μnk )∞ k=1 converges weakly to μ. Indeed, by tightness of  , for every r = 1, 2, . . . there exists a compact set Kr ⊂ S such that μn (Kr ) ≥ 1 − 1/r for every n ≥ 1. It is clear that Kr is also a compact subset of S and therefore, Kr ∈ β(S) ∩ β( S) and μnk (Kr ) = μnk (Kr ). But μnk ⇒ ν. Therefore, ν(Kr ) ≥ lim supk→∞ μnk (Kr ) ≥ 1 − 1/r. Thus E = : r≥1 Kr ⊂ S and E ∈ β(S) ∩ β( S). For every A ∈ β(S) we ∈ β( have A ∩ E ∈ β(S), because A ∩ E = A ∩ S ∩ E = A ∩ E for every A S). Put μ(A) = ν(A∩E) for every A ∈ β(S). It is clear that μ is a probability measure on (S, β(S)) and μ = ν. Finally, we verify that μnk ⇒ μ. Indeed, let A be a closed = μn (A). ∩ S for every closed set A ⊂ subset of S. Then A = A S and μn (A) Therefore, lim supk→∞ μnk (A) = lim supk→∞ μnk (A) ≤ μ(A) = μ(A) , which   by virtue of Theorem 1.5.1, implies that μnk ⇒ μ.

30

1 Preliminaries

Let (Xn )∞ n=1 be a sequence of S-random variables Xn : n → S on probability spaces (n , Fn , Pn ) for n ≥ 1. We say that (Xn )∞ n=1 is weakly convergent in distribution to a random variable X :  → S defined on a probability space (, F, P ) if the sequence (P Xn−1 )∞ n=1 of distributions of random variables Xn : n → S is weakly convergent to the distribution P X−1 of X. It is denoted by Xn ⇒ X. Similarly, the above sequence (Xn )∞ n=1 is said to be tight if a set {P Xn−1 : n ≥ 1} is a tight subset of the space M(S). If Xn and X are defined on the same probability, space (, F, P ), then we can define convergence of the above sequence (Xn )∞ n=1 in probability and a.s. to a random variable X. We denote P

the above types convergence by Xn → X and Xn → X a.s., respectively. Corollary 1.5.2 If (Xn )∞ n=1 and X are such as above, then Xn ⇒ X if and only if En {f (Xn )} → E{f (X)} as n → ∞ for every f ∈ Cb (S), where En and E are mean value operators taken with respect to probability measures Pn and P , respectively. Proof By the definition of weak convergence of sequencesof random variables and probability measures, it follows that Xn ⇒X if and only if S f (x) d [P (Xn )−1 ] →  −1 S f (x) d [P (X) ] as n → ∞ for every f ∈ Cb (S). The result follows now from the equalities S f (x) d [P (Xn )−1 ] = n f (Xn ) dPn = En {f (Xn )} and   −1   S f (x) d [P (X) ] =  f (X) dP = E{f (X)}. Lemma 1.5.3 Let (S, ρ) and (Y, d) be metric spaces, and Xn and X be Srandom variables defined on a probability space (n , Fn , Pn ) and (, F, P ), respectively, for n = 1, 2, . . . and such that Xn ⇒ X as n → ∞. For every continuous mapping  : S → Y one has  ◦ Xn ⇒  ◦ X as n → ∞. Proof By virtue of Theorem 1.5.1 for every open set G ⊂ S one has lim infn→∞ P (Xn )−1 (G) ≥ P X−1 (G). By continuity of  for every open set U ⊂ Y a set −1 (U ) is an open set of S. Taking in the above inequality G = −1 (U ), we obtain lim infn→∞ P (Xn )−1 (−1 (U )) ≥ P X−1 (−1 (U )). But P (Xn )−1 (−1 (U )) = Pn [(Xn )−1 (−1 (U ))] = P ( ◦ Xn )−1 (U ) and P X−1 (−1 (U )) = P [X−1 (−1 (U ))] = P ( ◦ X)−1 (U ) for every open set U ⊂ Y . Therefore, for every open set U ⊂ Y one has lim infn→∞ P ( ◦ Xn )−1 (U ) ≥ P ( ◦ X)−1 (U )], which by Theorem 1.5.1 and the definition of weak convergence of sequences of random variables, implies that  ◦ Xn ⇒  ◦ X as n → ∞.   We have the following result. Theorem 1.5.3 Let (S, ρ) be a Polish space, and (Pn )∞ n=1 be a sequence of M(S) weakly convergent to P ∈ M(S) as n → ∞. Then there exist a P , P ) and S-random variables Xn and X on ( ) , F, , F probability space ( −1 −1 for n = 1, 2, . . . such that : (i) Pn = P Xn for n = 1, 2, . . . , P = P X and (ii) ρ(Xn , X) → 0 a.s. as n → ∞.  

1.5 Random Variables

31

From Theorem 1.5.3 the following result follows. Corollary 1.5.3 Let (S, ρ) be a Polish space and (Xn )∞ n=1 be a sequence of random elements Xn : n → S, defined for every n ≥ 1 on probability spaces (n , Fn , Pn ), convergent in distribution to a random element X :  → S, defined on a P ) and , F, probability space (, F, P ). Then there exist a probability space ( n :  :  n−1 and → S and X → S such that P Xn−1 = P X random elements X −1 −1 = P X for n ≥ 1, and ρ(X n , X) → 0, P -a.s. as n → ∞. PX   In what follows, the above results will be considered with metric spaces : = S = : C([0, T ], Conv(Rd )), with the metric ρ defined by ρ(Z, Z) d sup0≤t≤T h(Z(t), Z(t)) for every Z, Z ∈ C([0, T ], Conv(R )), and S = :  −n z) = ∞ C(R+ , Rd ) with a metric ρd defined by ρd (z, n=1 2 {1 ∧ sup0≤t≤n |z(t) − z(t)|} for z, z ∈ C(R+ , Rd ). It is clear that (C(R+ , Rd ), ρd ) and (Conv(Rd ), h) are Polish spaces. Therefore, C([0, T ], Conv(Rd )) and C(R+ , Rd ) are complete and separable. For simplicity of notations, spaces C(R+ , Rd ) and Conv(Rd ) will be denoted by C(Rd ) and X , respectively. In further applications of the above results, we shall need necessary and sufficient conditions for the tightness of sequences of random variables with values in C([0, T ], X ) and C(Rd ), respectively. From the definition of the tightness of subsets of the space M(S) with and S = C(R+ , X ) and S = C(Rd ), it follows that such conditions are connected with necessary and sufficient conditions of relative compactness of subsets of the space C([0, T ], X ) and C(Rd ), respectively. In particular, for subsets of the space C([0, T ], X ) they follow from the Ascoli theorem presented above, and the following result dealing with the relatively compactness of subsets of the space (Comp(Rd ), h). Theorem 1.5.4 A subset K of the metric space (Comp(Rd ), h) is relatively compact if and only if it is bounded. Proof  Let K be a nonempty bounded subset of the space (Comp(Rd ), h), and let  = [ {A : A ∈ K}]. By the boundedness of the set K there exists a number M > 0 such that A ≤ M for every A ∈ K, where A = sup{|a| : a ∈ A}. Therefore, for every x ∈  one has |x| ≤ M. Thus,  is a bounded subset of Rd , which implies that (, λ) with λ(x, y) = |x − y| for every x, y ∈ Rd , is a compact metric space. It is clear that K ⊂ Comp(). By Theorem 1.3.3, every sequence ∞ (An )∞ n=1 ⊂ K possesses a subsequence (Ank )k=1 convergent to A ∈ Comp(). Thus, K is a relatively sequentially compact subset of a metric space (, λ), which is equivalent to its relatively compactness. Conversely, if K is relatively compact subset of the metric space (Comp(Rd ), h), then its closure K, taken with respect to the Hausdorff metric topology, is a compact subset of this space. Then K is a bounded subset of the space (Comp(Rd ), h). Therefore, K is a bounded, because K ⊂ K.   Corollary 1.5.4 A subset K of the metric space (X , h) is relatively compact if and only if it is bounded.

32

1 Preliminaries

Proof The result follows from Corollary 1.3.1, because the family X of all nonempty compact convex subsets of the space Rd is a compact subset of the space (Comp(Rd ), h).   Remark 1.5.1 From Corollary 1.5.4 it follows that for a given set H ⊂ C([0, T ], X ), a set H(t) = {z(t) : z ∈ H} is for every 0 ≤ t ≤ T a relatively compact subset of the space (X , h) if and only if it is a bounded subset of this space. Moreover, if a set H ⊂ C([0, T ], X ) is uniformly equicontinuous, then the boundedness of the set H(t) = {z(t) : z ∈ H} for every 0 ≤ t ≤ T is equivalent to the following condition (∗) : there exists M > 0 such that sup0≤t≤T supz∈H z(t) ≤ M. Indeed, it is obvious that (∗) implies boundedness of the set H(t) for every 0 ≤ t ≤ T . Conversely, suppose that a set H ⊂ C([0, T ], X ) is uniformly equicontinuous and H(t) = {z(t) : z ∈ H} is a bounded subset of the space (X , h) for every 0 ≤ t ≤ T . By uniform equicontinuity of the set H it follows that for every t ∈ [0, T ] and ε > 0 there exists δ(t, ε) > 0 such that for every z ∈ H one has h(z(t), z(s)) < ε for every s ∈ [0, T ] such that |t − s| < δ(t, ε). Let ε = 1. Then for every t ∈ [0, T ] there exists δ(t, 1) > 0 such that for every z ∈ H and s ∈ B(t, δ(t, 1)) one has h(z(t), z(s)) < ε, where B(t, δ(t, 1)) = {s ∈ [0, T ] : |t − s| < δ(t, 1)}. Hence, by boundedness of the set H(t) = {z(t) : z ∈ H} for every 0 ≤ t ≤ T , one gets z(s) ≤ h(z(t), z(s)) + z(t) ≤ 1 + Mt for every 0 ≤ t ≤ T , where Mt > 0 is such that H(t)  ≤ Mt for every 0 ≤ t ≤ T with H(t) = sup{|a| : a ∈ H(t)}. But [0, T ] ⊂ 0≤t≤T B(t, 1)). Then there exists a finite set {t1 , . . . , tn } ⊂ [0, T ] δ(t, n such that [0, T ] ⊂ B(t i , δ(ti , 1)). Therefore, for every s ∈ [0, T ] there i=1 exists i ∈ {1, . . . , n} such that s ∈ B(ti , δ(ti , 1)). Then for every s ∈ [0, T ] and z ∈ H one has z(s) ≤ 1 + max1≤i≤n Mti . Thus, the boundedness of the set H(t) = {z(t) : z ∈ H} for every 0 ≤ t ≤ T and the uniform equicontinuity of the set H, implies that the condition (∗) is satisfied.   Theorem 1.5.5 Let (Xn )∞ n=1 be a sequence of random elements Xn : n → C([0, T ], X ) on a probability space (n , Fn , Pn ) for n = 1, 2, . . .. Then (Xn )∞ n=1 is tight if and only if (i) limN →∞ supn≥1 Pn ({maxt∈[0,T ] Xn (t) > N }) = 0 and (ii) limδ↓0 supn≥1 Pn ({Vδ (Xn ) > ε}) = 0 for every ε > 0 where Vδ (Xn ) = maxt,s∈[0,T ],|t−s|≤δ h(Xn (t), Xn (s)). Proof By virtue of (i) for every ε > 0 there exists a number Nε > 0 such that P Xn−1 ({z ∈ C([0, T ], X ) : max0≤t≤T z(t) > Nε }) ≤ ε/2 for n ≥ 1. Then for every n ≥ 1 one has P Xn−1 ({z ∈ C([0, T ], X ) : max0≤t≤T z(t) ≤ Nε }) ≤ 1 − ε/2. By (ii), for every ε > 0 and k = 1, 2, . . . there exists δkε > 0 such that δkε ↓ 0 as k → ∞ and P Xn−1 ({z ∈ C([0, T ], X ) : Vδkε (z) > 1/k}) ≤ ε/2k+1  for every n ≥ 1. Therefore, we have P Xn−1 ( ∞ k=1 {z ∈ C([0, T ], X ) : Vδk (z) ≤ 1/k})  > 1 − ε/2. Taking Kε = {z ∈ C([0, T ], X ) : max0≤t≤T |z(t)| ≤ ε Nε } ∩ ∞ k=1 {z ∈ C([0, T ], X ) : Vδk (z) ≤ 1/k} we infer that for every z ∈ Kε and every 0 ≤ t ≤ T one has z(t) ≤ Nε and limk→∞ Vδkε (z) = 0. Therefore, by the Ascoli theorem, a set Kε is for every ε > 0 a relatively compact subset of the space C([0, T ], X ) such that P Xn−1 (Kε ) > 1 − ε for n ≥ 1. But Kε is for

1.5 Random Variables

33

every ε > 0 a closed subset of the space C([0, T ], X ). Therefore, it is compact for every ε > 0. Then for every ε > 0 there is a compact set Kε ⊂ C([0, T ], X ) such that P Xn−1 (Kε ) > 1 − ε for n ≥ 1. Thus, a sequence (Xn )∞ n=1 of random elements Xn : n → C([0, T ], X ) is tight. Conversely, let (Xn )∞ n=1 be tight. Then for every α > 0 there exists a compact set Kα ⊂ C([0, T ], X ) such that for every n ≥ 1 one has P Xn−1 (Kα ) > 1 − α. By the Ascoli theorem, a set Kα is uniformly equicontinuous and such that a set Kα (t) = {z(t) : z ∈ Kα } is a relatively compact subset of the space (X , h) for every 0 ≤ t ≤ T . Hence, by Corollary 1.5.4 and Remark 1.5.1, it follows that the condition (∗), presented in Remark 1.5.1, is satisfied. Thus, supz∈Kα sup0≤t≤T z(t) < ∞. Let N0 = supz∈Kα sup0≤t≤T z(t). Then Kα ⊂ {z ∈ C([0, T ], X ) : sup0≤t≤T z(t) ≤ N0 }, which implies that 1 − α ≤ Pn (Kα ) ≤ Pn ({z ∈ C([0, T ], X ) : sup0≤t≤T z(t) ≤ N0 }). Therefore, for every α > 0 and N ≥ N0 one has Pn ({z ∈ C([0, T ], X ) : sup0≤t≤T z(t) > N }) ≤ Pn ({z ∈ C([0, T ], X ) : sup0≤t≤T z(t) > N0 }) < α, which implies that Pn ({sup0≤t≤T Xn  > N }) ≤ α for every n ≥ 1 and N ≥ N0 . Then limN →∞ supn≥1 Pn ({sup0≤t≤T Xn  > N}) = 0. To prove condition (ii), let us note that by the uniform equicontinuity of the set Kα , it follows that for every ε > 0 and α > 0 there exists δ > 0 such that Kα ⊂ {z ∈ C([0, T ], X ) : Vδ (z) ≤ ε}, which implies that 1 − α ≤ Pn (Kα ) ≤ Pn ({z ∈ C([0, T ], X ) : Vδ (z) ≤ ε}). Hence, by the definition of the distribution P Xn−1 , it follows that P Xn−1 ({z ∈ C([0, T ], X ) : Vδ (z) ≤ ε}) ≥ 1 − α for every n ≥ 1. Therefore, for every n ≥ 1, ε > 0 and α > 0 one gets Pn ({Vδ (Xn ) > ε}) ≤ α, which implies that limδ↓0 supn≥1 Pn (Vδ (Xn ) > ε) = 0 for every ε > 0.   Theorem 1.5.6 Let (Xn )∞ n=1 be a sequence of random elements Xn : n → C([0, T ], X ) on probability spaces (n , Fn , Pn ) for n = 1, 2, . . . and such that and γ such that En [max0≤t≤T Xn (t)γ ] ≤ M (i) there exist positive numbers M for n = 1, 2, . . . , and (ii) there exist M > 0 and positive numbers α, β such that En [hα (Xn (t), Xn (s))] ≤ M|t − s|1+β for n = 1, 2, . . . and t, s ∈ [0, T ]. Then ∞ a sequence (Xn )∞ n=1 is tight, and there exists an increasing sequence (nk )k=1 of , P ) and random elements X nk :  , F → positive integers, a probability space ( C([0, T ], X ) and X :  → C([0, T ], X ) on ( , F , P ) for k = 1, 2, . . . , such n−1 for k = 1, 2, . . . , and such that ρ(X nk , X) → 0, P -a.s. as that P Xn−1 = PX k k k → ∞, where ρ is the supremum metric of the space C([0, T ], X ). γ Proof By Chebyshev’s inequality we get P ({max0≤t≤T Xn (t) > N}) ≤ M/N for n = 1, 2, . . .. Therefore, the condition (i) of Theorem 1.5.5 is satisfied. For simplicity we assume now that T > 0 is a positive integer. By (ii) the process Y = (Y (t))t≥0 defined by Y (t) = Xn (t) for a fixed n = 1, 2, . . . , satisfies En [hα (Y (t), Y (s))] ≤ M|t − s|1+β for t, s ∈ [0, T ]. Hence, by Chebyshev’s inequality, applied to every a > 0, it follows that Pn ({h(Y ((i + 1)/2m ), Y (i/2m ))>1/2ma })≤M2−m(1+β) 2maα =M2−m(1+β−aα) for i = 0, 1, 2, . . . , 2m T − 1. Taking now a number a such that 0 0 and δ > 0 and select ν = ν(δ, ε) such that (1 + 2/(2a − 1))/2aν ≤ ε ∞ −m(β−aα) < δ/T M . For such selected numbers ε > 0 δ > 0 and and T m=ν 2 ν = ν(ε, δ) we obtain Pn (

∞  m=ν

{

max m

0≤i≤2 T −1

∞ 

h(Y ((i + 1)/2m ), Y (i/2m )) > 1/2ma })≤M

2−m(β−aα) 1/2ma } and ∼ ν = n \ ν . We have Pn (ν ) < δ and if ω ∈ ν , then h(Y ((i + 1)/2m ), Y (i/2m )) ≤ 1/2ma for m ≥ ν and all i = 0, 1, 2 . . . such that (i + 1)/2m ≤ T . Let DT be a set of all dyadic rational numbers of [0, T ]. If s ∈ DT ∩ [i/2ν , (i + 1)/2ν ), then j it can be expressed by the formula s = i/2ν + l=1 αl /2ν+1 with αl ∈ {0, 1}. Therefore, for such s on the set ∼ ν , one has h(Y (s), Y (i/2 )) ≤ ν

j 

 h Y (i/2 + ν

k=1 j 

αl /2(ν+k)a ≤

k=1

k 

ν+l

αl /2

), Y (i/2 + ν

l=1 ∞ 

k−1 

 ν+l

αl /2

) ≤

l=1

αl /2(ν+k)a = 1/(2a − 1)2aν .

k=1

ν Therefore, on ∼ ν for every s, t ∈ DT satisfying |s − t| ≤ 1/2 we get

h(Y (s), Y (t)) ≤ 1 +

2 /2aν ≤ ε. 2a − 1

Indeed, if t ∈ [(i − 1)/2ν , i/2ν ) and s ∈ [i/2ν , (i + 1)/2ν ) , then h(Y (s), Y (t)) ≤ h(Y (s), Y (i/2ν )) + h(Y (t), Y ((i − 1)/2ν ))+

h(Y (i/2ν ), Y ((i − 1)/2ν )) ≤ 1 +

2 /2aν . 2a − 1

If t, s ∈ [i/2ν , (i + 1)/2ν ), then h(Y (s), Y (t)) ≤ h(Y (s), Y (i/2ν )) + h(Y (t), Y (i/2ν )) ≤

(2a

2 . − 1)2aν

But DT is dense in [0, T ] and h(Y (s), Y (t)) ≤ ε for every s, t ∈ DT . Then Pn ({maxt,s∈[0,T ],|t−s|≤1/2ν h(Y (s), Y (t)) > ε}) ≤ Pn (ν ) < δ for every s, t ∈ [0, T ] satisfying |s − t| ≤ 1/2ν . But ν = ν(δ, ε) does not depend on

1.5 Random Variables

35

n. Therefore, this implies that the condition (ii) of Theorem 1.5.5 is also satisfied. −1 Thus, a sequence (Xn )∞ n=1 is tight. Let Pn = P Xn for every n ≥ 1. By the ∞ n )∞ is a tight definition of tightness of the sequence (Xn )n=1 , it follows that (P n=1 sequence of the space M(S) with S = C([0, T ], X ). Therefore, by Theorem 1.5.2 nk )∞ of (P n )∞ and a probability measure P ∈ M(S) there is a subsequence (P k=1 n=1 as k → ∞. By virtue of Theorem 1.5.3 there exist a nk ⇒ P such that P , P ) and random elements X nk :  , F → C([0, T ], X ) and probability space ( , P : ) for k = 1, 2, . . . , such that P X n−1 = P Xn−1 → C([0, T ], X ) on (  , F X k k −1 = P for k = 1, 2, . . . , and ρ(X nk , X) → 0, P -a.s. as k → ∞. and P X   The following result can be proved in a similar way. Theorem 1.5.7 Let (ζn ) be a sequence of random variables ζn defined on a probability space (n , Fn , Pn ) with values in C([0, T ], Rm ). Assume the following conditions are satisfied : (i) limC→∞ limn→∞ sup0≤t≤T P ({|ζn (0)| > C}) = 0 and (ii) limδ→0 limn→∞ sups1 ,s2 ∈[0,T ],|s1 −s2 |≤δ P ({|ζn (s1 ) − ζn (s2 )| > ε}) = 0 for every P ), an increasing sequence , F, ε > 0. Then there exist a probability space ( ζnk ))k≥1 of random variables ζnk and a (nk )k≥1 of positive integers, a sequence ( P ) with values in C([0, T ], Rm ) such that , F, random variable ζ , defined on ( - a.s. for P ζn−1 =P ζn−1 for k ≥ 1 and such that sup0≤t≤T | ζnk (t) − ζ (t)| → 0 P k k every 0 ≤ t ≤ T as k → ∞.   with values in a separable metric space For given random elements Z and Z P ), respectively, we shall , F, (S, λ), defined on probability space (, F, P ) and ( d −1 . In what follows, we shall consider S to be the write Z = Z if P Z −1 = P Z Cartesian product C([0, T ], Rm ) × C([0, T ], X ). It will be denoted by S(Rm , X ) and considered as a metric space with a metric D defined by D[(x, X), (y, Y )] = sup0≤t≤T |x(t) − y(t)| + ρ(X, Y ) for every (x, X), (y, Y ) ∈ S(Rm , X ). From Theorem 1.5.6 and Theorem 1.5.7 the following result follows. Corollary 1.5.5 If the assumptions of Theorems 1.5.6 and 1.5.7 are satisfied, then there are an increasing sequence (nk )∞ k=1 of positive integers, a probability P ), random elements X nk :  :  , F, → C([0, T ], X ), X → space ( → C([0, T ], Rm ) and → C([0, T ], Rm ), such that C([0, T ], X ), ζn k :  ζ : d nk ) for k = 1, 2, . . . , and D[( nk ), ( → 0, P ζn k , X ζn k , X ζ , X)] (ζnk , Xnk ) = ( a.s. as k → ∞.   In a similar way (see [29], Th. 2.7, Th. 4.2 of Chap. I) the following results for C(Rd )-random variables can be proved. d Theorem 1.5.8 Let (Xn )∞ n=1 be a sequence of C(R )-random variables Xn on a probability space (n , Fn , Pn ) such that : (i) limN →∞ supn≥1 Pn ({|Xn (0)| > N }) = 0 and (ii) limh↓0 supn≥1 Pn ({maxt,s∈[0,T ],|t−s|≤h |Xn (t)−Xn (s)| > ε}) = 0 for every T > 0 and ε > 0. Then there exists an increasing sequence (nk )∞ k=1 , P ) and C(Rd )-random variables , F of positive integers, a probability space (

36

1 Preliminaries

, P for k = 1, 2, . . . on (  ) such that P Xn−1 = P X n−1 for k = nk and X , F X k k nk , X) → 0 a.s. as k → ∞. 1, 2, . . . and ρd (X   d n Theorem 1.5.9 Let (Xn )∞ n=1 be a sequence of C(R )-random variables X on probability spaces (n , Fn , Pn ) for n = 1, 2, . . . such that there exist positive numbers M and γ such that : (i) En [|Xn (0)|γ ] ≤ M for n = 1, 2, . . . , and for every T > 0 there exist MT > 0 and positive numbers α, β independent on T > 0 such that : (ii) En [|Xtn − Xsn |α ] ≤ MT |t − s|1+β for n = 1, 2, . . . and t, s ∈ [0, T ]. Then (Xn )∞   n=1 satisfies conditions (i) and (ii) of Theorem 1.5.8.

Similarly as above we consider the Cartesian product S(Rm , Rd ) = C(R+ , Rm ) × C(R+ , Rd ) with a metric d defined by d[(x, X), (y, Y )] = ρm (x, y) + ρd (X, Y ) for every (x, X), (y, Y ) ∈ S(Rm , Rd ), where ρm and ρd are metrics on C(R+ , Rm ) and C(R+ , Rd ), respectively. From Theorem 1.5.9 the following result follows. ∞ Corollary 1.5.6 Let (Xn )∞ n=1 and (ζn )n=1 be sequences of random elements Xn : d m n → C(R ) and ζn : n → C(R ) on probability spaces (n , Fn , Pn ) for n = 1, 2, . . . satisfying the assumptions of Theorem 1.5.9, and such that for every T > 0 and ε > 0 one has : (i) limN →∞ supn≥1 Pn ({|ζn (0)| > N}) = 0, and (ii) limh↓0 supn≥1 Pn ({maxt,s∈[0,T ],|t−s|≤h |ζn (t) − ζn (s)| > ε}) = 0. Then there are an increasing sequence (nk )∞ k=1 of positive integers, a probability space (, F , P ), nk , X and C(Rm )-random elements ζ nk , ζ , defined C(Rd )-random elements X d nk nk , P ) such that (ζ nk , Xnk ) = , F on ( (ζ , X ) for k = 1, 2, . . . , and such that n n k k , X) → 0, P - a.s. as k → ∞. ζ , ζ ) + ρd (X   ρm (

Given a filtered probability space PF = (, F, F, P ), a random variable T :  → [0, ∞] such that {T ≤ t} ∈ Ft for every t ≥ 0 is said to be an F-stopping time. As usual {T ≤ t} = {ω ∈  : T (ω) ≤ t}. In some cases the condition {T ≤ t} ∈ Ft can be replaced by {T < t} ∈ Ft for every t ≥ 0. This follows from the following theorem. Theorem 1.5.10 If a filtered probability space PF is such that F is right continuous, then a random variable T :  → [0, ∞] is an F-stopping time on PF if and only if {T < t} ∈ Ft for every t ≥ 0.  Proof Let {T < t} ∈ Fu for u > t and t ≥ 0. Since {T ≤ t} = t+ε>u>t {T < u} for every ε  > 0 and the filtration F = (Ft )t≥0 is right continuous, then we have {T ≤ t} ∈ u>t Fu = Ft for t ≥ 0. Therefore, the condition {T < t} ∈ Ft for t ≥ 0 implies that {T ≤ t} ∈  Ft for t ≥ 0. Conversely, if {T ≤ t} ∈ Ft for t ≥ 0, then we also have {T < t} = ε∈Q s∈Q∩[0,t−ε] {T ≤ s} ∈ Ft , where Q is the set of all rational numbers of the real line R.  

1.6 Stochastic Processes

37

1.6 Stochastic Processes From the practical point of view, random variables can be applied to mathematical modeling of static random processes. In the case of dynamic processes, instead of random variables we have to applied families X = (Xt )t≥0 of random variables parametrized by a parameter t ≥ 0, usually treated as the time at which the modeled dynamical process is taking place. A family X = (Xt )t≥0 of n-dimensional random variables Xt :  → Rn is called n-dimensional stochastic process on a probability space (, F, P ). It is clear that the above-defined stochastic process X = (Xt )t≥0 can be treated as a mapping X : R+ ×  → Rn such that X(t, ·) is a random variable for every t ≥ 0. Therefore, the stochastic process X = (Xt )t≥0 can be also denoted as a family (X(t, ·))t≥0 of sections X(t, ·) at t ≥ 0 of the mapping X : R+ ×  → Rn . A process X = (Xt )t≥0 is called continuous if for a.e. ω ∈  mappings R+  t → Xt (ω) ∈ Rn , called trajectories of X, are continuous. In a similar way we define càdlàg and càglàd stochastic processes on PF . In particular, a process X = (Xt )t≥0 is said to be càdlàg if for a.e. ω ∈  its trajectory R+  t → Xt (ω) ∈ Rn is right continuous and possesses the left-hand limit Xt− (ω) for every t > 0. Similarly, a process X = (Xt )t≥0 is called a càglàd if for a.e. ω ∈  its trajectory R+  t → Xt (ω) ∈ Rn is left continuous and possesses the righthand limit Xt+ (ω) for every t > 0. A stochastic process X = (Xt )t≥0 , defined on a filtered probability space PF = (, F, F, P ) with a filtration F = (Ft )t≥0 , is called F-adapted if a random variable Xt is Ft -measurable, for every t ≥ 0. The process X = (Xt )t≥0 is said to be measurable if a mapping X : R+ ×  → Rr defined by X(t, ω) = Xt (ω) for (t, ω) ∈ R+ ×  is (β(R+ ) ⊗ F, β(Rr ))-measurable. It is called F-non-anticipative if it is measurable and F-adapted. A process X = (Xt )t≥0 is said to be F-progressively measurable if for all t ≥ 0 its restriction to It ×  with It = [0, t] is (β(It ) ⊗ Ft , β(Rr ))-measurable. We can consider F-predictable and F-optional processes. Such processes are measurable with respect to a predictable σ -algebra P(F) generated by all F-adapted càglàd processes on PF and an optional σ -algebra O(F) generated by all F-adapted càdlàg processes on PF , respectively. It can be verified that P(F) ⊂ O(F) ⊂ β(R+ ) ⊗ F. Therefore, each predictable process is optional and both are measurable. It is clear that every F-progressively measurable process is F-non-anticipative. Similarly as above, for a given family G = {Xλ : λ ∈ } of d-dimensional stochastic processes, we can define an σ algebra generated by a family G. It is denoted by σ (G) and defined to be the smallest σ -algebra such that Xλ is (σ (G), β(Rr ))-measurable for every λ ∈ . We have the following Kolmogorov’s extension theorem. Theorem 1.6.1 Let μt1 ,...,tk be for all t1 , . . . , tk ∈ [0, ∞) and k ∈ N the probability measure on β(Rkd ) such that (i) μtσ (1) ,...,tσ (k) (Aσ (1) × . . . × Aσ (k) ) = μt1 ,...,tk (A1 ×. . .×Ak ) for all permutations σ = (σ (1), . . . , σ (k)) of {1, 2, . . . , k} and (ii) μt1 ,...,tk (A1 × . . . × Ak ) = μt1 ,...,tk ,tk+1 ,...,tk+m (A1 × . . . × Ak × d R . . × Rd) for all m ∈ N. Then there exist a probability space (, F, P )  × . m

and a d-dimensional stochastic process X = (Xt )t≥0 on (, F, P ) such that

38

1 Preliminaries

μt1 ,...,tk (A1 × . . . × Ak ) = P ({Xt1 ∈ A1 , . . . , Xtk ∈ Ak }) for ti ∈ [0, ∞) and Ai ∈ β(Rd ) with i = 1, 2, . . . , k and k ∈ N.   Stochastic processes X = (Xt )t≥0 and Y = (Yt )t≥0 , defined on a probability space (, F, P ) are called indistinguishable if P ({Xt = Yt : t ≥ 0}) = 1. A process Y is said to be a modification of X if P ({Xt = Yt }) = 1 for every t ≥ 0. The properties of the above type “equivalency” of two stochastic processes are quite different. If X and Y are modifications, then for every t ≥ 0 there exists a null set t ⊂  such that if ω ∈  t , then Xt (ω) = Yt (ω). Since the interval [0, ∞) is uncountable the set  = t≥0 t could have any probability between 0 and 1, and it could be even non-measurable. If X and Y are indistinguishable, then there exists a null set  ⊂  such that if ω ∈ , then Xt (ω) = Yt (ω) for all t ≥ 0. In other words, the paths of X and Y are the same for all ω ∈ . But if processes X and Y are right continuous, then the above defined “equivalency” are the same. It follows from the following theorem. Theorem 1.6.2 Let X = (Xt )t≥0 and Y = (Yt )t≥0 be two stochastic processes, with X a modification of Y . If X and Y are right continuous, then they are indistinguishable. Proof Let 0 ⊂  be such that all paths of X and Y corresponding to ω ∈  \ 0 + are Let t = {Xt = Yt } and  =  right continuous on R and P (0 ) = 0. + . We have P () = 0 and P ( ∪  , where Q denotes the rationals in R 0 t∈Q t ) = 0. Then Xt (ω) = Yt (ω) for t ∈ Q and ω ∈ 0 ∪ . For fixed t ∈ R+ we can select a sequence (tn )∞ n=1 of Q such that tn → t as n → ∞. We can assume that tn decrease to t through Q. Then we get Xt (ω) = limn→∞ Xtn (ω) = limn→∞ Ytn (ω) = Yt (ω) for ω ∈ 0 ∪  and every t ≥ 0.   We have the following existence continuous modification theorem. Theorem 1.6.3 Suppose a d-dimensional stochastic process X = (Xt )t≥0 on PF is such that for all T > 0 there exist positive constants α, β , and γ such that E|Xt − Xs |α ≤ γ |t − s|1+β for s, t ∈ [0, T ]. Then there exists a continuous modification of the process X.   Let us note that for a given stochastic process X = (Xt )t≥0 on PF we may identify each ω ∈  with its path R+  t → Xt (ω) ∈ Rd . Thus we may regard  = (Rd )[0,∞) of all functions from [0, ∞) into Rd . Then as a subset of the space  the σ -algebra F will contain the σ -algebra B, generated by sets {ω ∈  : ω(t1 ) ∈ A1 , . . . , ω(tk ) ∈ Ak } for all t1 , . . . , tk ∈ R+ and all Borel sets Ai ⊂ Rr for i = 1, 2, . . . , k and k ∈ N. The space (Rd )[0,∞) contains among others spaces like that : C(Rd ) = : C(R+ , Rd ), C(Rd )+= : C+ (R+ , Rd ) and C(Rd )−= : C− (R+ , Rd ) of all continuous bounded, right continuous bounded, and left continuous bounded, respectively, functions x : R+ → Rd . A special role in such approach to stochastic processes plays an evaluation mapping defined for every fixed t ≥ 0 by setting et : (Rd )[0,∞)  x → x(t) ∈ Rd . We can define on the space S = (Rd )[0,∞) an σ -algebra of cylindrical sets, denoted by C(S), as a σ -algebra generated by a family

1.6 Stochastic Processes

39

{et : t ≥ 0}, i.e., C(S) = σ ({et : t ≥ 0}). In a similar way we can define a filtration (Ct )t≥0 by taking Ct = σ ({es : 0 ≤ s ≤ t}). We have the following result. Theorem 1.6.4 The σ -algebra C(C(Rd )) of cylindrical sets of C(Rd ) coincides with the σ -algebra β(C(Rd )) of Borel sets of C(Rd ). Proof It is only to verify that β(C(Rd )) ⊂ C(C(Rd )). Let us observe that a family of sets {x ∈ C(Rd ) : max0≤t≤n |x(t) − x0 (t)| ≤ ε} with fixed x0 ∈ C(Rd ), ε > 0 and n = 1, 2, . . . is a base of neighborhoods in C(Rd). On the other hand, we have {x ∈ C(Rd ) : max0≤t≤n |x(t) − x0 (t)| ≤ ε} = r∈Q,0≤r≤n {x ∈ C(Rd ) : x(r) ∈ U (x0 (r), ε)}, where U (a, ε) = {x ∈ Rr : |x − a| ≤ ε}. Therefore, {x ∈ C(Rd ) : max0≤t≤n |x(t)−x0 (t)| ≤ ε} ∈ C(C(Rd )), which implies that β(C(Rd )) ⊂ C(C(Rd )).   , P ) and random , F Corollary 1.6.1 Given probability spaces (, F, P ) and ( :  −1 if → C(Rd ), we have P X−1 = P X elements X :  → C(Rd ) and X −1 −1 n for every n ≥ 1, where Yn = (Xt1 , Xt2 , . . . , Xtn ) and and only if P Yn = P Y n = (X t1 , X t2 , . . . , X tn ) for every t1 < t2 < . . . < tn < ∞. Y   Given a d-dimensional continuous stochastic process X = (Xt )t≥0 defined on a probability space (, F, P ) its distribution, denoted by P X−1 is defined as the distribution of C(Rd )-random element X :  → C(Rd ). Similarly, the distribution of càdlàg process is defined. From Theorems 1.5.8 and 1.5.9 the following result follows. Theorem 1.6.5 Let (Xn )∞ n=1 be a sequence of d-dimensional continuous stochastic processes Xn = (Xtn )t≥0 on probability spaces (n , Fn , Pn ) such that : (i) there exist positive numbers M and γ such that En [|X0n |γ ] ≤ M for n = 1, 2, . . . and (ii) for every T > 0 there exist MT > 0 and positive numbers α, β independent of T > 0 such that En [|Xtn − Xsn |α ] ≤ MT |t − s|1+β for n = 1, 2, . . . and t, s ∈ [0, T ]. Then there exists an increasing sequence (nk )∞ k=1 , P ), d-dimensional stochastic , F of positive integers, a probability space ( , P nk = (X tnk )t≥0 and X = (X t )t≥0 on (  ) for k = 1, 2, . . . , F processes X n −1 n −1 k k nk , X) → 0 a.s. and such that P (X ) = P (X ) for k = 1, 2, . . . , and d(X as k → ∞.   There are two important classes of stochastic processes: martingales and Markov processes. We characterize them by giving their most important properties. Similarly as above let PF be a filtered probability space (, F, F, P ) with a filtration F = (Ft )t≥0 satisfying the usual conditions. A real-valued F-adapted stochastic process X = (Xt )t≥0 is said to be an F-martingale or simply martingale (supermartingale, submartingale) on PF if (i) E|Xt | < ∞ for t ≥ 0 and (ii) E[Xt |Fs ] = Xs (E[Xt |Fs ] ≤ Xs , E[Xt |Fs ] ≥ Xs ) a.s. for 0 ≤ s ≤ t < ∞. A martingale X = (Xt )t≥0 on PF is called closed by a random variable Y on PF if E|Y | < ∞ and Xt = E[Y |Ft ] a.s. for t ≥ 0. Given a n-dimensional F-adapted process on PF is said to be an F-Markov process if for every 0 ≤ s ≤ t < ∞ and every bounded Borel measurable function f : Rn → R one has E[f (Xt )|Fs ] =

40

1 Preliminaries

E[f (Xt )|σ (Xs )] a.s. By the above definitions it follows that the martingale property means that for a given present time s, the process has no tendency in future times t ≥ s, that is, the average over all future possible states of Xt gives just the present state Xs . In the difference to this, the Markov property means that the present has no memory, that is, the average of Xt knowing the past is the same as the average of Xt knowing the present. The following results will be applied in some later estimations. Theorem 1.6.6 (Jensen’s Inequality) Assume that ϕ : R → R is convex, and let X and ϕ(X) be integrable random variables. For every σ -algebra G ⊂ F one has ϕ (E[X|G]) ≤ E[ϕ(X)|G].   Theorem 1.6.7 (Doob’s Inequality) Let X = (Xt )0≤t≤T be a right continuous submartingale. Then for every λ > 0 one has P ({ sup Xt ≥ λ}) ≤ 0≤t≤T

1 λ

 {sup0≤t≤T Xt ≥λ}

XT dP ≤

1 E|XT |. λ

(1.6.1)

If X is a right continuous non-negative submartingale such that E|XT2 | < ∞, then   E(sup0≤t≤T Xt2 ) ≤ 4E(XT2 ). Corollary 1.6.2 Let X = (Xt )t≥0 be a martingale and ϕ : R → R be convex such that ϕ(Xt ) is integrable for 0 ≤ t < ∞. Then ϕ(X) is a submartingale. In particular, |X| and X2 are submartingales.   An F-adapted càdlàg process X = (Xt )t≥0 on PF is said to be a local Fmartingale if there exists an increasing sequence (Tn )∞ n=1 of F-stopping times Tn with limn→∞ Tn = ∞ a.s. such that the process (1{Tn >0} Xt∧Tn )t≥0 is a uniformly integrable martingale for each n ≥ 1, where similarly as above 1{Tn >0} denotes the characteristic function of the set {Tn > 0} = {ω ∈  : Tn (ω) > 0}. Such sequence (Tn )∞ n=1 of F-stopping times is called a fundamental sequence of the local martingale X. It can be verified that if X and Y are continuous real-valued local martingales, then there exists a unique (up to the indistinguishability) F-adapted continuous process of bounded variation X, Y with X, Y 0 = 0 a.s. such that XY − X, Y is a continuous local martingale. The process X, Y is called the cross-variation of X and Y . If X = Y we write X = X, X and called this process the quadratic variation of X. It is non-decreasing and F-adapted. In many applications we have to deal with stochastic processes X = (Xt )t≥0 which are representable (not necessarily in a unique way) as sums X = X0 +A+M, where A = (At )t≥0 is a càdlàg, F-adapted process with paths of finite variation on compacts and M = (Mt )t≥0 is a local F-martingale on a given filtered probability space PF = (, F, F, P ) with a filtration F = (Ft )t≥0 satisfying the usual conditions. Such processes are said to be semimartingales on PF . We shall define now a special process an a filtered probability space PF , known as F-Brownian motion, that is, an F-martingale and an F-Markov process. An mdimensional F-adapted process B = (Bt )t≥0 on PF is called an m-dimensional

1.6 Stochastic Processes

41

F-Brownian motion or a Brownian motion on PF if (i) for every 0 ≤ s ≤ t < ∞, Bt − Bs is independent of Fs and (ii) for every 0 ≤ s ≤ t < ∞, Bt − Bs is a Gaussian random variable with mean zero and variance matrix (t − s)C for a given non-random matrix C. The Brownian motion B = (Bt )t≥0 is said to start at x ∈ Rr if P ({B0 = x}) = 1. The existence of an F-Brownian motion can be proved using a path-space construction, together with the Kolmogorov’s extension theorem. But it is not true that on every complete filtered probability space PF there exists a Brownian motion. Sometimes the underlying probability space PF is just too small. Nevertheless, it can be proved that there exists a complete filtered probability space such that there exists a Brownian motion on this space. If B = (Bt )t≥0 is a realvalued F-Brownian motion such that B0 = 0 a.s., then the stochastic process M = (Mt )t≥0 defined by Mt = Bt2 − t a.s. for every t ≥ 0 is an F-Brownian motion. It is not hard to verify (see [83], Exercise 2.8 of Chap. 2) that an m-dimensional FBrownian motion B = (Bt )t≥0 satisfies E|Bt − Bs |4 ≤ m(m + 2)|t − s|2 for every 0 ≤ s < t < ∞. Then by Theorem 1.6.3, it possesses a continuous modification. In what follows, we shall always assume that we have to deal with a continuous modification of each m-dimensional F-Brownian motion B = (Bt )t≥0 . We shall assume that such modification possesses the matrix C equal to the identity matrix. We have the following properties of F-Brownian motions. Theorem 1.6.8 (i) For every α < 1/2 almost all paths of Brownian motions are Hölder continuous with exponent α. (ii) For any α > 1/2 almost all paths of Brownian motions are nowhere Hölder continuous with exponent α.   Theorem 1.6.9 Let X = (Xt )t≥0 be an m-dimensional continuous, F-adapted j process on PF such that (i) E[Xt − Xs |Fs ] = 0 a.s. and (ii) E[(Xti − Xsi )(Xt − j Xs )] = δij (t − s) a.s. for every 0 ≤ s ≤ t < ∞, where δij = 0 for i = j and δij = 1 for i = j . Then X is an F-Brownian motion on PF .   Corollary 1.6.3 (i) Almost all sample paths of a Brownian motion are nowhere differentiable. (ii) Almost all sample paths of a Brownian motion have infinite variation on any finite interval.   Remark 1.6.1 A real Brownian motion can be defined on some probability space (, F, P ) as a continuous stochastic process β = (βt )t≥0 such that β0 = 0 and β is a stationary process with independent Gaussian increments such that E[βt − βs ] = 0 and E[(βt − βs )2 ] = σ 2 (t − s) for every 0 ≤ s < t < ∞. In such β β case we can define a filtration Fβ = (Ft )t≥0 with an augmented σ -algebra Ft defined for every t ≥ 0 by a family {βs : 0 ≤ s ≤ t} of random variables, i.e.,  β β β Ft = s>t σ (Fs ∪ N ), where Ft = σ {βs : s ≤ t} and N is the collection of all P -null sets in F. It can be verified that β is a real Fβ -Brownian motion on a β filtered probability space (, F, Fβ , P ). It can be proved that the filtration (Ft )t≥0  β β β β β β is continuous, i.e., that Ft− = Ft = Ft+ , where Ft− = σ ( st σ (Fs ) ,and F0− = F0 .

42

1 Preliminaries

Remark 1.6.2 In what follows, we shall apply Corollaries 1.5.5 and 1.5.6 with a sequence (ζn ) defined by ζn = B n , where B n = (Btn )t≥ is for every n ≥ 1 an mdimensional Brownian motion on a probability space (n , Fn , Pn ). Then the results where B is an m-dimensional of these Corollaries will be satisfied with ζ = B, Brownian motion on a probability space (, F, P ).   Let us denote by D(F, Rd ) the space of all d-dimensional F-adapted càdlàg processes X = (Xt )t≥0 on PF and let Db (F, Rd ) be its subset containing all bounded elements of D(F, Rd ). As it was mentioned above, we can define on R+ ×  the predictably σ -algebra P(F) generated by D(F, Rr ). By Pb (F) we denote a σ -algebra on R+ ×  generated by Db (F, Rd ). Finally, by S(F, Rd ) we denote the space of all continuous d-dimensional F-martingales M = (Mt )t≥0 such that | M | 2c = E[supt≥0 |Mt |2 ] < ∞. It can be verified that (S(F, Rd ), | · | c ) is a Banach space.

1.7 Properties of Exit Times of Continuous Processes Let D be a domain in Rd and (s, x) ∈ R+ × D . Assume that X = (X(·, t))t≥0 and Xn = (Xn (t, ·))t≥0 are continuous stochastic processes on a probability space (, F, P ) such that X(s, ·) = Xn (s, ·) = x a.s. for n = 1, 2, . . . and such that supt≥0 |Xn (t, ·) − X(t, ·)| → 0 a.s. as n → ∞. Let τ = inf{r > s : X(r, ·) ∈ D} and τn = inf{r > s : Xn (r, ·) ∈ D} for n = 1, 2, . . . . Random variables τ and τn are said to be exit times form the set D of processes X and Xn , respectively. The following result (see [48], Th. 5.1 of Chap. 4) can be proved. Theorem 1.7.1 If a set D, a sequence (Xn )∞ n=1 and a process X are such as above, then τn → τ a.s. as n → ∞.   In the stochastic optimal control theory the following result is applied. Theorem 1.7.2 Let D be a domain in Rd and (s, x) ∈ R+ × D. Assume X = = (X(t)) t≥0 are continuous d-dimensional stochastic processes (X(t))t≥0 and X P ) , respectively, such that X(s) = x a.s., and P X−1 = on (, F, P ) and (, F, −1 −1 ◦ P X . Then P (τD ) = P ( τD )−1 , P (X ◦τD )−1 = P (X τD )−1 , and P (τD , X ◦ −1 −1 τD ) = P ( τD , X ◦ τD ) , where τD = inf{t > s : Xt ∈ D} and τD = inf{t > t ∈ D} . s:X Proof Let η : C(Rd ) → R+ be defined by η(x) = inf{t > s : x(t) ∈ D} for x ∈ C(Rd ) . It is clear that η is (β, β+ )-measurable, where β+ denotes the Borel σ -algebra on R+ . Taking Y = R+ , G = β+ and  = η , we get τD =  ◦ X . Therefore, by Lemma 1.5.2 we obtain P (τD )−1 = P ( and τD =  ◦ X τD )−1 . d + Let ψ(t, x) = x(t) for x ∈ C(R ) and t ∈ R and put (x) = ψ(η(x), x)) for x ∈ C(Rd ) . It is clear that the mapping  satisfies conditions of Lemma 1.5.2 with Y = Rd and G = β , where β denotes the Borel σ -algebra on Rd . Furthermore, = X◦ τD . Therefore, by virtue of Lemma 1.5.2, we have ◦X = X◦τD and ◦ X

1.8 Stochastic Integrals

43

◦ we obtain P (X ◦ τD )−1 = P (X τD )−1 . Finally, let (x) = (η(x), ψ(η(x), x)) d for x ∈ C(R ) . From the properties of the mappings ψ and η, it follows that  satisfies the conditions of Lemma 1.5.2 with Y = R+ × Rd and G = β+ × β , where β+ denotes the Borel σ -algebra of R+ . Furthermore, ◦X = (τD , X◦τD ) = ( ◦ and ◦ X τD , X τD ) , which by Lemma 1.5.2 implies that P (τD , X ◦τD )−1 = −1 ◦ τD ) .   P ( τD , X Corollary 1.7.1 If the assumptions of Theorem 1.7.2 are satisfied, then for every continuous bounded function f : R+ × Rd → R one has E[f (τD , X ◦ τD )] = ( ◦ denote the mean value operators with respect E[f τD , X τD )] , where E and E to probability measures P and P , respectively. It is also true if the function f : ( ◦ R+ × Rd → R is Borel measurable and E[f (τD , X ◦ τD )] < ∞ and E[f τD , X τD )] < ∞.

1.8 Stochastic Integrals Classical analysis has its disposal various approaches to the operation of integration, which bring about such concepts as Riemann, Riemann–Stieltjes, Lebesgue, Bochner, and other integrals. In stochastic analysis one also considers various approaches to integration of random functions with respect to stochastic processes, stochastic measures, and so on, which brings about various constructions of stochastic integrals. For instance, stochastic integration with respect to finite variation of stochastic processes can be thought of as an extension of path-by-path Lebesgue–Stieltjes integration. Unfortunately, in practical applications we have to deal with processes with almost all paths of infinite variations on compacts. The most important example of such processes is the Brownian motion. Therefore, it was important to define stochastic integrals in another way than the Lebesgue– one. N. Wiener was the first to define (see [85, 97]) the stochastic integral Stieltjes t + → R using the idea f (τ )dB τ for smooth deterministic function f : R 0 of “integration by parts” of the form d(f · B) = f dB + Bdf and got the t t t formula 0 f (τ )dBτ = f (t)Bt − 0 f  (τ )Bτ dτ, where the integral 0 f  (τ )Bτ dτ is treated as trajectory-wise Riemann integral of the continuous function R+  t → f  (t)Bt (ω) for a.e. fixed ω ∈ . K. Itô made in 1944 a significant step forward (see [30]) in the extension of the concept of a stochastic integral and laid on this way the foundations of modern stochastic calculus. Later on, Itô’s approach has been extended on the case of stochastic integrals with respect to martingales and semimartingales. We recall the main ideas of these approaches and the basic properties of such defined stochastic integrals. We begin with the Itô stochastic integrals of an F-non-anticipative stochastic processes with respect to a Brownian motion defined on a given filtered probability space PF = (, F, F, P ) with a filtration F = (Ft )t≥0 satisfying the usual conditions. To present Itô’s approach and basic properties of the Itô integrals, let us denote by M2F (a, b) the family of restrictions of all F-non-anticipative real-valued processes X = (Xt )t≥0 to the

44

1 Preliminaries

b interval [a, b] such that P ({ a |Xt |2 dt < ∞}) = 1. By L2F (a, b) we denote a b subset of M2F (a, b) of all X ∈ M2F (a, b) such that E[ a |Xt |2 dt] < ∞. A stochastic process X ∈ M2F (a, b) is called simple if there exists a partition a = t0 < t1 < . . . < tr = b of [a, b] such that Xt = Xti for ti ≤ t < ti+1 with i = 0, 1, . . . , r − 2 and Xt = Xtr−1 for tr−1 ≤ t ≤ b. The class of all simple processes of M2F (a, b) is denoted by SF (a, b). Every F ∈ SF (a, b) can be presented by r−2 1[ti ,ti+1 ) ϕi + 1[tr−1 ,b] ϕr−1 , where ϕi is an Fti -measurable real random F = i=0 variable on PF for i = 0, 1, . . . , r −1. It can be proved that for every f ∈ M2F (a, b) b P n there exists a sequence (f n )∞ n=1 ⊂ SF (a, b) such that a |ft − ft |dt → 0 as b n → ∞. In particular, if f ∈ L2F (a, b), then E a |ft − ftn |2 dt → 0. Let B = (Bt )t≥0 be a one-dimensional F-Brownian motion on PF such that Ba = 0. By Itô stochastic integral of F ∈ SF (a, b) with respect to a Brownian b motion B we mean a real random variable on PF , denoted by a Ft dBt , and defined b r−1 by setting a Ft dBt = Ft -measurable i=0 ϕi (Bti +1 − Bti ), where ϕi are the r−2 i random variables for i = 0, 1, . . . , r − 1 such that F = i=0 1[ti ,ti+1 ) ϕi + 1[tr−1 ,b] ϕr−1 . From the above definition the basing properties of Itô integrals define on the space SF (a, b) follows. Lemma 1.8.1 Let F, F,1 F 2 ∈ SF (a, b), λ1 , λ2 ∈ R, ε > 0 and N > 0. Then b b b (a) a (λ1 Ft1 + λ2 Ft2 )dBt = λ1 a Ft1 dBt + λ2 a Ft2 dBt a.s., b (b) if F ∈ SF (a, b) ∩ L2F (a, b), then E[ a Ft dBt ] = 0, b b (c) P ({| a Ft dBt | > ε}) ≤ P ({ a |Ft |2 dt > N}) + N/ε2 , b b (d) if F ∈ SF (a, b) ∩ L2F (a, b), then E| a Ft dBt |2 = E a |Ft |2 dt. Proof Conditions (a), (b), and (d) follow from the definition of the Itô stochastic integral. Indeed, for F ∈ SF (a, b)∩L2F (a, b), From the above definition Itô integral it follows that   E 

b

a

r−1 

2 r−1   Ft dBt  = E[|Fti |2 (Bti+1 − Bti )2 ] = i=0

E|Fti |2 · E[(Bti+1 − Bti )2 ] =

i=0

r−1 

 E|Fti |2 (ti+1 − ti ) = E

i=0

b

|Ft |2 dt .

a

For the proof of (c), let N (t) be defined by N (t) =

⎧ ⎨ Ft

if

⎩0

if

tk ≤ t < tk+1 and tk ≤ t < tk+1 and

k

j =0 |Ftj |

k

2 (t

j =0 |Ftj |

j +1

2 (t

j +1

− tj ) ≤ N − tj ) > N

for k = 0, 1, 2, . . . , r − 1, where a = t0 < t1 < . . . < tr = b. The process b N = (N (t))a≤t≤b belongs to SF (a, b) ∩ L2F (a, b), and a |N (t)|2 dt =

1.8 Stochastic Integrals

ν

45

 − tj ), where ν is the largest integer such that kj =0 |Ftj |2 (tj +1 − b tj ) ≤ N , ν ≤ r − 1. Hence it follows that E a |N (t)|2 dt ≤ N . Further, b Ft − N (t) = 0 for all t ∈ [a, b] if a |N (t)|2 dt < N. Therefore, j =0 |Ftj |

   P 

b a

2 (t

j +1

 

    ≤P Ft dBt  > ε 

a

b

 

  N (t)dBt  > ε +P

b

 |Ft |2 dt > N

.

a

By Chebyshev’s inequality, the first integral on the right-hand side is bounded by b (1/ε2 )E| a N (t)dBt |2 ≤ N/ε2 . Therefore, (c) is satisfied.   To extend the above definition of stochastic integrals on the whole space M2F (a, b), we need the following result. Lemma 1.8.2 For every F ∈ M2F (a, b) and every sequence (F n )∞ n=1 of SF (a, b) b n P such that a |Ft − Ft |2 dt → 0 as n → ∞, there exists a real random variable J (F ) on PF , independent of the particular choice of the sequence (F n )∞ n=1 , such b n P that a Ft dBt → J (F ) as n → ∞. b n P 2 Proof Let (F n )∞ n=1 be a sequence of SF (a, b) such that a |Ft − Ft | dt → 0 as b P n → ∞. Hence it follows that a |Ftn − Ftm |2 dt → 0 as n, m → ∞. By virtue of Lemma 1.8.1, for every ε > 0 and ρ > 0, we get

   P 

b a

 Ftn dBt

b

− a

 

Ftm dBt 

 >ε



b

≤ ρ+P a

 |Ftn

− Ftm |2 dt

2

>ε ρ

.

b Then the sequence ( a Ftn dBt )∞ n=1 is a Cauchy sequence with respect to the convergence in probability. By completeness with respect to convergence in probability of the space of all square integrable real random variables on PF , there is a random b P variable J (F ) on PF such that a Ftn dBt → J (F ) as n → ∞. b n P 2 Suppose (Gn )∞ n=1 is a sequence of SF (a, b) such that a |Gt − Ft | dt → 0 as 2n = F n and H 2n+1 = Gn satisfies n → ∞. The sequence (H n )∞ n=1 defined by H b n P 2 it follows that there exists the random a |Ht − Ft | dt → 0 as n → ∞. Hence b n variable K(F ) such that the sequence ( a Ht dBt )∞ n=1 converges in probability to  b 2n ∞ K(F ). Therefore, its subsequence ( a Ht dBt )n=1 also converges in probability to   K(F ). By the definition of H 2n it follows that J (F ) = K(F ) a.s. b The random variable J (F ) defined in Lemma 1.8.2 is denoted by a Ft dBt and said to be the Itô integral of F ∈ M2F (a, b) with respect to the F-Brownian motion b B = (Bt )t≥0 . In particular, a Ft dBt ∈ L2 (, F, R) for F ∈ L2F (a, b). We have the following properties of stochastic integrals defined above.

46

1 Preliminaries

Theorem 1.8.1 Let F , F,1 F 2 ∈ M2F (a, b), λ1 , λ2 ∈ R, ε > 0 and N > 0. Then the following relations are satisfied b b b (a) a (λ1 Ft1 + λ2 Ft2 )dBt = λ1 a Ft1 dBt + λ2 a Ft2 dBt a.s., b b (b) P ({| a Ft dBt | > ε}) ≤ P ({ a |Ft |2 dt > N}) + N/ε2 . Proof The equality (a) is a consequence of the definition of the Itô integral and Lemma 1.8.1. For the proof of (b), let us assume that (F n )∞ of n=1 is a sequence b n b 2 SF (a, b) such that limn→∞ a |Ft − Ft | dt = 0 a.s. By the definition of a Ft dBt b P b we have a Ftn dBt → a Ft dBt as n → ∞. By virtue of Lemma 1.8.1, we have P

   

b a

 

  ≤P Ftn dBt  > ε

a

b

|Ftn |2 dt > N 

 +

N (ε )2

for ε > ε and N < N  . Passing to the limit by n → ∞, using the above property b   of the sequence ( a Ftn dBt )∞ n=1 , and taking ε ↑ ε, and N ↓ N , we obtain

   P 

a

b

 

  ≤P Ft dBt  > ε

b

 |Ft |2 dt > N

+

a

N . ε2  

2 Theorem 1.8.2 Let F ∈ M2F (a, b), and let (F n )∞ n=1 be a sequence of MF (a, b) b n   P P b b such that a |Ft − Ft |2 dt → 0 as n → ∞. Then a Ftn dBt → a Ft dBt as n → ∞.

Proof By Theorem 1.8.1 for every ε > 0 and ρ > 0 one has  



 b   n   ≤ρ+P P  (Ft − Ft )dBt  > ε a

b a

 |Ftn

− Ft | dt > ε ρ 2

2

for n = 1, 2, . . .. From this and properties of the sequence (F n )∞ n=1 , the result follows.   b b 2 2 Theorem 1.8.3 If F ∈ LF (a, b), then (i) E a Ft dBt = 0 and (ii) E| a Ft dBt | = b E a |Ft |2 dt.  2 (a, b) such that E b |F n − Proof Let (F n )∞ n=1 be a sequence of SF (a, b) ∩ LF b  ba t Ft |2 dt → 0 as n → ∞. This implies that E a |Ftn |2 dt → E a |Ft |2 dt b as n → ∞. By virtue of Lemma 1.8.1, we get E a Ftn dBt = 0 and b b E| a Ftn dBt |2 = E a |Ftn |2 dt for every n = 1, 2, . . .. Hence in particular, b 2 it follows that ( a F n dBt )∞ n=1 is a Cauchy sequence of L (, F, P , R). By b virtue of Theorem 1.8.2, it converges in probability to a Ft dBt , which implies

1.8 Stochastic Integrals

47

b b b that E| a Ftn dBt |2 → E| a Ft dBt |2 as n → ∞. Then E| a Ft dBt |2 = b n   b b limn→∞ E| a Ft dBt |2 = limn→∞ E a |Ftn |2 dt = E a |Ft |2 dt.   Corollary 1.8.1 For every G ∈ L2 ([a, b] × , F , Rr ), F ∈ M2F (a, b), and b b b ψ ∈ L2 (, Fa , R), one has E a (ψ ·G)t dt = E[ψ a Gt dt] and a (ψ ·F )t dBt = b ψ a Ft dBt . Proof It is clear that ψ · G ∈ L2 ([a, b] × , F , R), and ψ · F ∈ M2F (a, b). From b Fubini’s theorem and properties of the integral a (ψ · G)t dt one obtains  E

b

 (ψ · G)t dt = E

a

b



 (ψ · G)t dt = E ψ

a



b

Gt dt .

a

b n P 2 Let (F n )∞ t | dt → 0 as n → n=1 be a sequence of SF (a, b) such that a |Ft − F  b ∞. It is clear that ψ · F n ∈ SF (a, b) for every n ≥ 1 and a |(ψ · F n )t − (ψ · b b P F )t |2 dt → 0, because a |(ψ · F n )t − (ψ · F )t |2 dt = ψ 2 a |Ftn − Ft |2 dt and b b n P b P 2 Therefore, a (ψ · F n )t dBt → a (ψ · F )t dBt as a |Ft − Ft | dt → 0 as n → ∞. b b n → ∞. From the definition of a (ψ · F n )t dBt it follows that a (ψ · F n )t dBt = b b P b ψ a Ftn dBt . Furthermore we have a Ftn dBt → a Ft dBt as n → ∞. Therefore, b b P ψ a Ftn dBt → ψ a Ft dBt as n → ∞. But, for every n ≥ 1, we have   b  b    (ψ · F )t dBt − ψ Ft dBt  ≤  a a    b  b  b   (ψ · F )t dBt − (ψ · F n )t |dBt  + ψ Ftn dBt − ψ

    a a a  b    b     (ψ · F n )t dBt −  (ψ · F )t |dBt  + |ψ|   a

a

a

  Ft dBt  = a   b b  n Ft dBt − Ft dBt  . b

a

  b b P  b  Therefore,  a (ψ · F )t dBt − ψ a Ft dBt  = 0 a.s., because a (ψ · F n )t dBt → b n b P b   a (ψ · F )t dBt and a Ft dBt → a Ft dBt as n → ∞. Given the above filtered probability space PF , by L2F we shall denote the space of all F-non-anticipative processes f = (ft )t≥0 such that f ∈ L2F (0, T ) for every T > 0. For f ∈ L2F and a one-dimensional Brownian motion B = (Bt )t≥0 on PF a t stochastic process ( 0 fτ dBτ )t≥0 is called an indefinite integral corresponding to a pair (f, B). Corollary 1.8.2 For the pair (f, B), given above, the indefinite integral t ( 0 fτ dBτ )t≥0 is F-adapted.

48

1 Preliminaries

Proof Let T > 0 and suppose f ∈ SF (0, T ) ∩ L2F (0, T ). For every t ∈ [0, T ] t k−1 one has 0 fτ dBτ = i=1 fti (Bti+1 − Bti ), where tk = t. Hence it follows t that 0 fτ dBτ is Ft -measurable, because fti (Bti+1 − Bti ) is Ftk -measurable for of SF (0, T ) ∩ L2F (0, T ) such that i = 1, 2, . . . , k − 1. If (f n )∞ n=1 is a sequence t n t n 2 E 0 |ft − ft | dt → 0 as n → ∞, then 0 fτ dBτ is Ft -measurable for every fixed t t ∈ [0, T ] and n ≥ 1. Hence it follows that 0 fτ dBτ is Ft -measurable for every t t 0 ≤ t ≤ T and every T > 0, because 0 fτnk dBτ → 0 fτ dBτ a.s. as k → ∞ for every increasing sequence (nk )∞   k=1 of positive integers. Theorem 1.8.4 For every T > 0 and f ∈ L2F , there exists a continuous t modification (J (t))0≤t≤T of ( 0 fτ dBτ )0≤t≤T . Proof Let (f n )∞ (0, T ) ∩ L2F (0, T ) such that f n = n=1 be a sequence of S  Ft n k−2 n n 2 i=1 ϕi1[ti ,ti+1 ) + ϕk−1 1[tk−1,T ] and E 0 |ft − ft | dt → 0 as n → ∞. Put t t In (t) = 0 fτn dBτ and I (t) = 0 fτ dBτ for t ∈ [0, T ]. From the definition of In (t) s t t it follows that for every 0 ≤ s < t ≤ T one has 0 fτn dBτ − 0 fτn dBτ = s fτn dBτ a.s. Hence continuity of In = (In (t))0≤t≤T for every n = 1, 2, . . . follows. Furthermore, for every 0 ≤ s < t ≤ T and n = 1, 2, . . ., one has

 E[In (t)|Fs ] = E 

s

0

fτn dBτ

+E

0

 s≤tjn ε}) ≤ 2 E[|In (T )−Im (T )|2 ] = 2 E ε ε 0≤t≤T



T 0

|ftn −ftm |2 dt,

which by the properties of the sequence (f n )∞ n=1 , implies that P ({sup0≤t≤T |Ink+1 (t) −Ink (t)| > 2−k }) ≤ 2−k for every k = 1, 2, . . . and every increasing sequence (nk )∞ k=1 of positive integers. By Borel–Cantelli lemma, we obtain P ({ sup |Ink+1 (t) − Ink (t)| > 2−k for infinity many k}) = 0. 0≤t≤T

1.8 Stochastic Integrals

49

Therefore, for a.e. ω ∈  there exists k1 (ω) such that sup0≤t≤T |Ink+1 (t) − Ink (t)| > 2−k for k ≤ k1 (ω). Then the sequence (Ink (t))∞ k=1 is uniformly convergent for t ∈ [0, T ] a.s. Let J = (J (t)0≤t≤T be an a.s. limit of the sequence (Ink )∞ k=1 of continuous processes Ink = (Ink (t))0≤t≤T . It is a continuous stochastic process on PF . Since Ink (t) → I (t) for every t ∈ [0, T ] as k → ∞ in the L2 -norm topology, then we must have I (t) = J (t) a.s. for all t ∈ [0, T ].    t Corollary 1.8.3 For every T > 0 and f ∈ L2F , the process I = ( 0 fτ dBτ )0≤t≤T is an F-martingale and P ({ sup |I (t)| ≥ λ}) ≤ 0≤t≤T

for every λ > 0, where I (t) =

t 0

1 E λ2



T

|ft |2 dt

(1.8.1)

0

fτ dBτ .

Proof We can assume that I is a continuous process. For every n = 1, 2, . . . let In be the stochastic processes defined in the proof of Theorem 1.8.4. It is an Fmartingale. Therefore, by Doob’s inequality it follows that there exists an increasing 2 sequence (nk )∞ k=1 of positive integers such that Ink (t) → I (t) in the L -norm topology for all t ∈ [0, T ] as k → ∞. Then the process I = (I (t)0≤t≤T is also an Fmartingale. The inequality (1.8.1) now follows immediately from Doob’s inequality.   From the above results it follows that for every T > 0 and f ∈ L2F the process t I = ( 0 fτ dBτ )0≤t≤T is a continuous F-martingale such that E|I (t)|2 < ∞, for 0 ≤ t ≤ T . This is not true in the general case for f  ∈ M2F (0, ∞). But, t it can be verified that in such case the process 0 fτ dBτ is a local F0≤t≤T

martingale. It is enough to define  t for every n = 1, 2, . . . an F-stopping time Tn by setting Tn = inf{t > 0 : 0 |fτ |2 dτ ≥ n} ∧ n. Then P ({Tn ≤ n}) = 1, P ({Tn ≤ Tn+1 }) = 1 and P ({limn→∞ Tn = ∞}) = 1. For every n = 1, 2, . . . we  t∧T t ∞ have I (t ∧ Tn ) = 0 n fτ dBτ = 0 1{τ ≤Tn } fτ dBτ , and 0 E[1{τ ≤Tn } |fτ |2 ]dτ = n 2 0 E[1{τ ≤Tn } |fτ | ]dτ ≤ n. Then by [48, Th. 4.3 of Chap. 1] the process {I (t ∧Tn ) : t ≥ 0} is a square integrable F-martingale for every n = 1, 2, . . ., because for every n = 1, 2, . . . a family {I (t ∧ Tn ) : t ≥ 0} of random variables is uniformly integrable. Let us note that the above-defined Itô integral can be defined for F-nonanticipative matrix-valued processes with respect to vector-valued F-Brownian motions B = (B 1 , . . . , B m ), where B 1 , . . . , B m denote real-valued F-Brownian motions on PF such that B i and B j are independent for i = j . In such a case, we consider a matrix-valued stochastic process F = (f ij )n×m with f ij ∈ M2F (0, ∞) t and define for every 0 ≤ s < t < ∞ a multidimensional integral s Fτ dBτ to be an (n × 1) matrix of the form

50

1 Preliminaries



t s

⎛ ⎞∗ m  t m  t   nj j Fτ dBτ = : ⎝ fτ1j dBτj , . . . , ft dBt ⎠ , j =1 s

j =1 s

where x ∗ denotes the transpose of x ∈ Rn . It can be verified that all properties of the integral presented above can be extended to the multidimensional case. In particular, t it can be verified that an n-dimensional stochastic process ( 0 Fτ dBτ )t≥0 possesses t a continuous modification. In what follows, ( 0 Fτ dBτ )t≥0 will be considered as its continuous version. In the theory of stochastic processes we have no differentiation theory in the classical sense, only an integration theory. Nevertheless, it turns out that it is possible to establish an integral version of the chain rule, called Itô’s formula. It is very useful for applications and concerns to Itô stochastic processes, called for simplicity Itô processes. Let PF = (, F, F, P ) with F = (Ft )t≥0 satisfying the usual conditions and B = (B 1 , . . . , B m ) be an m-dimensional F-Brownian motion on PF . Assume F = (f 1 , . . . , f n )∗ and G = (g ij )n×m are F-non-anticipative processes with f i ∞  ∞ ij and g ij such that P ({ 0 |fti |dt < ∞}) = 1 and P ({ 0 |gt |2 dt < ∞}) = 1 for i = 1, 2, . . . , n and j = 1, 2, . . . , m. The stochastic process  t n-dimensional t X = (Xt )t≥0 on PF defined by Xt = X0 + s Fτ dτ + s Gτ dBτ a.s. for t ≥ 0 is said to be a n-dimensional Itô process starting at X0 with the stochastic differential dX on [0, ∞) denoted by dXt = Ft dt + Gt dBt for t ≥ 0. We have the following theorem known as Itô’s lemma. Theorem 1.8.5 Let X = (Xt )t≥0 be a n-dimensional Itô process on PF having a stochastic differential dXt = Ft dt + Gt dBt for t ≥ 0 with F = (f 1 , . . . , f n )∗ and G = (g ij )n×m such as above. Assume g : [0, ∞) × Rn → Rp is an C 1,2 -map. Then the process Y = (Yt )t≥0 defined by Yt = g(t, Xt ) for t ≥ 0 is an p-dimensional Itô p process having a stochastic differential dY = (dYt )t≥0 with dYt = (dYt1 , . . . , dYt ) and dYtk defined by  ∂gk ∂gk 1   ∂ 2 gk j (t, Xt )dt + (t, Xt )dXti + (t, Xt )dXti dXt , ∂t ∂xi 2 ∂xi ∂xj n

dYtk =

n

n

i=1 j =1

i=1

for k = 1, 2, . . . , p, where i, j = 1, 2, . . . , m.

j dBti dBt

= δij dt, and dBti dt = dtdBti = 0 for  

Itô’s formula can be applied to some estimations of stochastic processes, and to solving some stochastic differential equations. Example 1.8.1 Let f = (ft )0≤t≤T be a real F-non-anticipative stochastic process T such that 0 E[ft2m ]dt < ∞. Then for every 0 ≤ t ≤ T one has

 E

2m

t

fτ dBτ 0



t

≤ [m(2m − 1)]m t m−1 0

E[fτ2m ]dτ.

(1.8.2)

1.8 Stochastic Integrals

51

t Indeed, let xt = 0 fτ dBτ and τN = inf{t : sup0≤s≤t |xs | ≥ N } with τN = T if sup0≤s≤t |xs | < N. Taking g(t, x) = x 2m for (t, x) ∈ [0, T ] × R, by Itô’s formula we get d[(xt∧τN )2m ] = 2m(xt∧τN )2m−1 d[xt ] + m(m − 1)(xt∧τN )2m−2 (d[xt ])2 , which can be written in the form  t∧τN  t∧τN 2m 2m−1 = 2m x f dB + m(2m − 1) xτ2m−2 fτ2 dτ. xt∧τ τ τ τ N 0

0

 2m ] = m(2m − 1)E t∧τN x 2m−2 f 2 dτ , which implies that a Hence it follows E[xt∧τ τ τ 0 N 2m ] ∈ R+ is non-decreasing. By the H¨ function [0, T ]  t → E[xt∧τ older inequality N with p = m and q = m/(m − 1) it follows that 

t∧τN

E 0

 t 1/m 2m m−1)/m 2m E xτ2m−2 fτ2 dτ ≤ t (m−1)/m (E[xt∧τ ]) f dτ . τ N 0

Thus,

 t 1/m 2m (m−1)/m 2m m−1)/m 2m E[xt∧τ E ] ≤ m(2m − 1)t (E[x ]) f dτ . t∧τN τ N 0

2m ] < ∞. Therefore, the above inequality is equivalent to But E[xt∧τ N t 2m 1/m 2m ] ≤ ≤ m(2m − 1)t (m−1)/m (E 0 fτ2m dτ )1/m . Then E[xt∧τ (E[xt∧τN ]) N  t 2m m m−1 E 0 fτ dτ . Hence, by the Fatou’s lemma it follows that (1.8.2) [m(2m − 1)] t is satisfied.  

Example 1.8.2 Let r, α ∈ R and X = (Xt )t≥0 be a stochastic process on PF such that dXt = r Xt dt + α Xt dBt for t ≥ 0, where B = (Bt )t≥0 is a given F-Brownian motion on PF . Using Itô’s formula we can determine the process X . To do this let us rewrite the above equation to the form dXt /Xt = rdt + αdBt . Taking g(t, x) = ln(x) for x > 0 , immediately from the Itô formula we obtain d(ln(Xt )) =



1 1 1 dXt 1 dXt 1 2 − 2 (dXt )2 = ·dXt + − 2 α 2 Xt2 dt = − α dt . Xt 2 Xt 2Xt Xt 2 Xt

Therefore,  rt + αBt =

t

0

dXt = Xt

 0

t

1 d(ln(Xs )) + α 2 t . 2

Assuming that X0 = 0 a.s. we get

ln

Xt X0



1 = (r − α 2 )t + αBt 2

52

1 Preliminaries

a.s. for t ≥ 0 . Then 1 Xt = X0 exp[(r − α 2 )t + αBt ] 2

a.s.

Thus X = (Xt )t≥0 is defined by Xt = X0 exp(μt + αBt ) a.s. for t ≥ 0 with   μ = (r − 12 α 2 ) . A process X = (Xt )t≥0 of the form Xt = X0 exp(μt + αBt ) with α, μ ∈ R is called geometric Brownian motions. Such processes are important as models for stochastic prices in mathematical economics. From the properties of stochastic processes defined by indefinite integrals it follows that for a given matrix-valued process G = (g ij )n×m with g ij ∈ M2F (0, ∞) and an m-dimensional F-Brownian motion B = (Bt )t≥0 , the process t X = (Xt )t≥1 with Xt = X0 + 0 Gτ dBτ for t ≥ 0 is a continuous n-dimensional local F-martingale. It can be proved that for martingales certain types the converse is also true. We precede the presentation of such a theorem by notions dealing with an extensions of filtered probability space. Given a filtered probability space PF = (, F, F, P ) with a filtration F = (Ft )t≥0 we will say that a filtered = ( t )t≥0 is an extension of PF if ) , with , F, probability space P F, P F = (F F F)-measurable mapping π :  t →  such that π −1 (Ft ) ⊂ F there exists an (F, ◦ π −1 and for every Z ∈ L∞ (, F, P , Rd ) an Rd for t ≥ 0 , P = P defined by setting Z( on P ω) = Z(π( random variable Z ω)) for ω ∈  F t ]( Z| F . There is a more satisfies E[ ω) = E[Z|]Ft ](π( ω)) for every ω ∈  general extension, called the standard extension, of a probability space PF . It is connected with the following problem: having given an F-adapted stochastic process X = (Xt )t≥0 on a filtered probability space PF = (, F, F, P ) with a filtration F = (Ft )t≥0 we may need an m-dimensional F-Brownian motion independent of X. But, because PF may not be rich enough to support such a Brownian motion, we must extend the probability space in order to construct this. To do this suppose ( , F  , P  ) is an another probability space on which we are given = =  ×  , F an m-dimensional Brownian motion B  = (B  )t≥0 , and let  t )t≥0 = P × P  , π( . If F ⊗ F , P ω) = ω for ω = (ω, ω ) ∈  F = (F , P t ⊂ Ft ⊗ { , ∅}, then ) such that Ft ⊗ F  ⊂ F , F is a filtration on ( = ( t )t≥0 is called a standard extension of the filtered ) with P , F, F, P F = ( F F probability space PF . It can be verified that a standard extension of the filtered probability space PF is an extension of this space. Let us observe that the filtration F defined above may not satisfy theusual conditions, so we augment it and make it t = right continuous by defining F s>t σ (Fs ∪ N ) , where N is the collection of . We also complete F by defining F = σ (F ∪ N ) . We may all P-null sets in F by defining X t ( extend X and B to F-adapted processes on P ω) = Xt (ω) and = (B t )t≥0 is an m-dimensional t ( . Then B ω) = Bt (ω ) for ω = (ω, ω ) ∈  B = (X t )t≥0 . Brownian motion, independent of X We can now formulate the following representation theorem.

1.8 Stochastic Integrals

53

Theorem 1.8.6 Suppose M = (M 1 , . . . , M d ) , with M i = (Mti )t≥0 for i = 1, 2, . . . , d , is a d-dimensional continuous local F-martingale on! PF such that for every i, j = 1, 2, . . . , d the function R+  t → M i , M j t (ω) ∈ R is = absolutely continuous for a.e. ω ∈ . Then there are a standard extension P F (, F , F, P ) of the space PF , a d-dimensional F-Brownian motion B = (Bt )t≥0 , and a matrix-valued process ρ = (ρ ij )d×d with ρ ij ∈ M2 (0, ∞) for on P F F i, j = 1, 2, . . . , d such that t τ for t ≥ 0 , (a) Mt = 0 ρτ d B t !  jk i j (b) M , M t = nk=1 0 ρτik ρτ dτ a.s. for t ≥ 0 and i, j = 1, 2, . . . , d .   We present now the general remarks dealing with the extension of the definition of the Itô stochastic integral to the case of integrations with respect to martingales and semimartingales. Let X = (Xt )t≥0 be a real-valued F-semimartigale defined on a filtered probability space PF = (, F, F, P ) with the filtration F = (Ft )t≥0 satisfying the usual conditions. Denote by E the class of simple processes on PF that are linear combinations of finitely many elementary processes fi (t, ω) = 1(τi−1 ,τi ] Zi (ω) with Fti−1 - measurable random variable Zi , i.e., every f ∈ E is  defined by f (t, ω) = 10 (t)Z0 (ω) + N 1(τi−1 ,τi ] Zi (ω) for (tω) ∈ [s, t] × .  i=1 t Similarly as above a stochastic integral 0 f (τ, ω)dXτ of the above simple process f t ∈ E with respect Nto a semimartingale X on the interval [0, t] is defined by f (τ, ω)dX = τ i=1 Zi (ω)[Xτi−1 ∧t − Xτi ∧t ] for (t, ω) ∈ [0, t] × . In the 0 present definition of a stochastic integral we need not to assume that X is a semimartingale. However, the semimartingale assumption is essential if we are going to extend the definition of the above stochastic integral to the much general class of stochastic processes than E. The procedure of such extension depends both on properties of local martingales and integrand processes. For example, if X = M, where M = (Mt )t≥0 is a square integrable martingale, i.e., such that supt≥0 E|Mt |2 < ∞, and a process M = ( M, M t )t≥0 of the quadratic variation of M is absolutely continuous a.s. with respect to t ≥ 0, then the set E 2 is dense in the space of measurable stochastic processes f = (ft )t≥0 on  ∞ L (M) PF such that E 0 |ft (ω)|2 d M, M t < ∞. Thus, for every f ∈ E there exists a ∞ ⊂ E such that E 0 |ftn (ω) − ft (ω)|2 d M, M t → 0. Then a sequence (f n )∞  t n=1 is convergent in the norm topology of L2 (, F, Rd ) sequence ( 0 fτn (ω)dXτ )∞  t n=1 to a random variable 0 fτ (ω)dXτ ∈ L2 (, F, Rd ). If M is continuous a.s., then the set E of simple processes is dense in the space L2 (M), and the random variable   ω → fτ (ω) (ω) ∈ R is Fτ -measurable for each finite F-stopping time τ :  → R+ . Finally, if we do not additionally ask for the regularity of the trajectories of M , then the set E is dense in the space of F-predictable processes on PF ∞ satisfying a boundedness condition E 0 |ft (ω)|2 d M, M t < ∞. Let us note that if we impose no regularity conditions on M , then the extension procedure of stochastic integrals can be applied for each predictable bounded processes f. t The next step in the extension of the definition of the stochastic integral 0 fτ dXτ

54

1 Preliminaries

is its extension to predictable locally bounded processes f and locally square integrable martingales, i.e., for processes f = (ft )t≥0 and M = (Mt )t≥0 such ∞ that there are sequences (τn )∞ n=1 and (κm )m=1 of F-stopping times “convergent” to τ n ∞, and such that stopped processes f = (ft∧τn )t≥0 and M κm = (Mt∧κm )t≥0 are bounded and square integrably bounded, respectively,  t for every n, m ≥ 1. The final step in the construction of stochastic integrals 0 fτ dXτ for local bounded predictable processes f = (ft )t≥0 with respect to F-semimartingale X = (Xt )t≥0 , is based on the following observation concerning the structure of semimartingales. If X = X0 +At + Mt for t ≥ 0, then A = (At )t≥0 is a process of bounded t variation, i.e., 0 |dAτ (ω)| < ∞, for t ≥ 0 and ω ∈ , and M = (Mt )t≥0 is an F-local martingale. But each local martingale M has a (not unique, in general) decomposition Mt = M0 + Mt + Mt for t ≥ 0, where M  = (Mt )t≥0 and M  = (Mt )t≥0 are local martingales M0 = M0 = 0 and M  has the bounded variation, while M  is locally square integrable. Hence each semimartingale X = (Xt )t≥0 can be representable as a sum Xt = X0 + At + Mt , where A = A + M  and M  is such as above. For locally  t bounded process f = (ft )t≥0 we have well-defined Lebesgue– Stieltjes integral 0 fτ (ω)dAτ for each ω ∈  and if in addition f is predictable, t then the stochastic integrals 0 fτ (ω)dMτ are also well defined. Thus, we can define t t t t the integral 0 fτ dXτ by setting 0 fτ dXτ = 0 fτ dAτ + 0 fτ dMτ . To show that this definition is consistent, we must to prove that for every two representations Xt = X0 + At + Mt , and Xt = X0 + A¯ t + M¯ t , one has  0

t

fτ dAτ +



t 0

fτ (ω)dMτ =



t 0

fτ d A¯ τ +



t

fτ d M¯ τ .

(1.8.3)

0

This is obvious for elementary processes f , and by linearity, holds also for simple functions. If f is predictable and bounded, then it can be approximated by simple processes f n convergent to f pointwise. Using this fact and localization procedure we obtain the required property (1.8.3). The last definition of stochastic integrals can be extended on the case of some F-predictable processes with respect to Fsemimartingale X. Similarly as in the case of stochastic integrals with respect to the Brownian motions, we can extend the definition of the above integrals on the case of vector-valued semimartingales and matrix-valued locally bounded predictable processes. The following results, dealing with stochastic integrals with respect to martingales, can be proved. Theorem 1.8.7 Let M = (Mt )t≥0 be an m-dimensional square integrable martingale on a filtered probability space PF = (, F, F, P ) with a filtration F = (Ft ) satisfying the usual conditions, and let φ be a measurable process such that φ ∈ L2 (M). Then for every fixed t > 0 one has

 t (a) E 0 φτ dMτ = 0 a.s., t s t (b) s φτ dMτ = 0 φτ dMτ − 0 φτ dMτ a.s., for every 0 ≤ s < t < ∞,

1.8 Stochastic Integrals

55

2   t t (c) E  0 φτ dMτ  = 0 E|φτ |2 d| M, M |τ ,  

 s t (d) E 0 φτ dMτ Fs = 0 φτ dMτ a.s., for every 0 ≤ s < t < ∞.

 

Apart from the quadratic variation process X = ( X, X )t≥0 of a continuous local martingale X = (Xt )t≥0 , defined above, we can define the quadratic variation process [X, X] = ([X, X]t )t≥0 of a semimartingale X = (Xt )t≥0 . It is defined by t setting [X, X]t = Xt2 − 2 0 X−τ dXτ . If X is such that [X, X] is locally integrable, then the conditional quadratic variation of X can be defined. It is still denoted by X, X , and understood as a finite variation natural process such that [X, X]− X, X is a local martingale. It can be proved that for every continuous semimartingale X we have [X, X] = X, X . In particular, for a standard Brownian motion B, we have [B, B]t = B, B t = t for all t ≥ 0. Given a semimartingales X = (Xt )t≥0 and Y = (Yt )t≥0 , the quadratic covariation [X,  t Y ] = ([X, Y ]t )t≥0 of X and Y is t defined by [X, Y ]t = (XY )t − 0 X−τ dXτ − 0 Y−τ dYτ . In particular, we have the following polarization identity : [X, Y ] = 12 ([X + Y, X + Y ] − [X, X] − [Y, Y ]). The conditional quadratic variation is also known by its notation. It is sometimes called the sharp bracket, the angle bracket, or the oblique bracket. It has properties analogous to that of the quadratic variation processes. In what follows, the following results will be needed. Theorem 1.8.8 (Kunita–Watanabe Inequality) Let M = (Mt )t≥0 and N = (Nt )t≥0 be m-dimensional square integrable martingales on a filtered probability space PF = (, F, F, P ) with a filtration F = (Ft ) satisfying the usual conditions, and let φ and ψ be measurable processes such that φ ∈ L2 (M) and ψ ∈ L2 (N ). Then 

t

 |φτ ||ψτ |d|[M, N ]τ | ≤

0

for every fixed t ≥ 0.

0

t

1/2   φτ2 d[M, M]τ

t 0

1/2 ψτ2 d[N, N ]τ  

= (B t )t≥0 be m-dimensional FTheorem 1.8.9 Let B = (Bt )t≥0 and B Brownian motions defined on a filtered probability space PF = (, F, F, P ) with a filtration F = (Ft ) satisfying the usual conditions. For every T > 0 and square d×m one has integrable F -non-anticipative t  t stochastic process f : [0, T ]× → R t 0 fτ dMτ = 0 fτ dBτ − 0 fτ d Bτ a.s. for every t ∈ [0, T ], where M = (Mt )t≥0 t a.s. for is a continuous square integrable F - martingale defined by Mt = Bt − B every t ≥ 0. Proof Let (f n )n≥1 be a sequence of simple square integrable F-non-anticipative stochastic processes f n : [0, T ] ×  → Rd×m of the form f n = T n ln −1 n 2 r=0 1[tr ,tr+1 ) φr + 1[tln −1 ,T ] φln −1 for n ≥ 1 and such that E 0 |fτ − fτ | dτ → 0 as n → ∞. Then

56

1 Preliminaries

 0 l n −1 r=0

t

fτn dMτ =

l n −1

φrn (Mtr+1 ∧t − Mtr ∧t ) =

r=0

φrn (Btr+1 ∧t − Btr ∧t ) −  0

l n −1

tr+1 ∧t − B tr ∧t ) = φrn (B

r=0 t



fτn dBτ

t

− 0

τ fτn d B

t t t τ for every n ≥ 1, which implies that L.i.m.( 0 fτn dMτ ) = 0 fτ dBτ − 0 fτ d B a.s. for every t ≥ 0, where L.i.m. denotes the limit with respect to the norm topology of the space L2 (, Ft , Rd ), as n → ∞. In order to finish the proof it is enough to verify that f is square integrable with respect to the martingale t M, i.e., that E 0 |fτ |2 d[M, M]τ < ∞. To do that let us note that [M, M]t = B − B] t = [B, B]t + [B, B] t − 2[B, B] [B− B, t  t t =2 2t − 2[B, B]t . Therefore, t 2 2 E 0 |fτ | d[M, M]t = 2E 0 |fτ | dτ − 2E 0 |fτ | d[B, B]τ . By the Kunita–  t τ | ≤ E t |fτ |2 dτ . Therefore, Watanabe inequality we have E 0 |fτ |2 |d[B, B] 0  t τ | ≤ E t |fτ |2 |d[B, B] τ | < ∞. Thus, E t |fτ |2 d[M, M]τ < | E 0 |fτ |2 d[B, B] 0 0 t ∞. In a similar way one can verify that E 0 |fτn − fτ |2 d[M, M]τ → 0 as t t n → ∞. Then by the Itô’s isometry L.i.m.( 0 fτn dMτ ) = 0 fτ dMτ . Therefore, t t t   0 fτ dMτ = 0 fτ dBτ − 0 fτ d Bτ . Finally, we shall present some general and auxiliary results dealing with convergence in distribution for stochastic integrals with semimartingale integrators. Let D(R+ , Rd ) denote a Polish metric space of all càdlàg functions provided with a Skorokhod topology (see [31], Chap. VI). Let Z = (Zt )t≥0 be an Fadapted semimartingale on a filtered probability space PF = (, F, F, P ) with a filtration F = (Ft )t≥0 satisfying the usual conditions, such that Z0 = 0. Let α : Rm → Rm be a bounded function with a compact support such that α(x) = x in some neighborhood of the origin. Then (see [91] and [31]) Z can be decomposed as Z = J α + N α + Aα , where N α is an F-adapted  local martingale, Aα is a predictable process with bounded variation, and J α = 0 0, there exists δ > 0 such that F (x) ¯ ⊂ V (F (x), ε) for every x ∈ B(x, ¯ δ), where V (F (x), ε) = {z ∈ X : dist(z, F (x)) ≤ ε} and B(x, ¯ δ) is an open ball of X centered at x¯ with radius δ. It can be verified that if F is h-l.s.c. at x¯ ∈ X, then it is also l.s.c. Indeed, if the multifunction F : X → P(Y ) is h-l.s.c. at x¯ ∈ X, then for every ε > 0 there exists a δ > 0 such that for every x ∈ X satisfying ρ(x, x) ¯ < δ one has ¯ (x), h(F ¯ F (x)) ≤ ε. Suppose the above conditions are satisfied and F is not l.s.c. at x. ¯ Then there exists an open set U ⊂ Y with F (x) ¯ ∩ U = ∅ and such that in every neighborhood V of x¯ there exists x ∈ V such that F ( x ) ∩ U = ∅. Therefore, we can select a sequence (xn )∞ of X convergent to x and such that F (xn ) ∩ U = ∅ n=1 for every n = 1, 2, . . .. On the other hand, for every ε > 0 there exists Nε ≥ 1 such that for every n ≥ Nε we have F (x) ¯ ⊂ V (F (xn ), ε). Hence in particular, it follows that F (x) ¯ ∩ U ⊂ V (F (xn ), ε) for n ≥ Nε . Let y ∈ F (x) ¯ ∩ U, nk = N1/k for every k = 1, 2, . . . and select yk ∈ F (xnk ) such that d(yk , y) < 1/k. For k sufficiently large we have yk ∈ U and therefore, F (xnk ) ∩ U = ∅. Contradiction. If F (x) ¯ ∈ Comp(Y ) and F is l.s.c. at x¯ ∈ X, then it is h-l.s.c. at x¯ ∈ X. Indeed, let F be l.s.c. at x¯ ∈ X and yi be for i = 1, . . . , m such that {B(yi , ε/2) : i = 1, . . . , m} covers F (x). ¯ Select for every i = 1, . . . , m a number δi > 0 such that ρ(x, x) ¯ < δi implies F (x) ∩ B(yi , ε/2) = ∅. Let δ = min{δi : i = 1, . . . , m}. Then ρ(x, x) ¯ 0, there ¯ (x), F (x)) exists δ > 0 such that h(F ¯ ≤ ε for every x ∈ B(x, ¯ δ). There are some h-u.s.c. multifunctions that are not u.s.c. This is illustrated in Figure 2.1, where F (t) = {(y, z) ∈ R2 : y = t} for t ∈ R. A set-valued mapping F : X → P(Y ) is said to be continuous (h-c) on X if it is l.s.c. (h-l.s.c.) and u.s.c. (h-u.s.c.) on X. It can be verified that a multifunction F : X → Comp(Y ) is continuous if and only if it is h-continuous. If (Y, | · |) is a Banach space and F : X → Conv(Y ), then F is continuous if and only if a function X  x → s(y ∗ , F (x)) ∈ R is continuous for every y ∗ ∈ Y ∗ , where s(·, A) denotes the support function of a set A ⊂ Y . In control theory, we have to deal with parameterized set-valued functions of the form F (x) = {f (x, u) : u ∈ U }, where f : X × U → Y is a given function. Some properties of such type multifunctions can be found in [48, Lemma 2.1 of Chap. 7]. We shall verify here only the following simple result. Lemma 2.1.1 Assume X and Y are topological Hausdorff spaces and let f : X × U → Y , where U = ∅. If f (·, u) is continuous on X for every u ∈ U , then the set-valued mapping F : X → P(Y ) defined by F (x) = f (x, U ) is l.s.c. on X.

2.1 Continuity of Multifunctions

63

Fig. 2.1 The mapping h-u.s.c. but not u.s.c. at t = 0

Proof Let x¯ ∈ X be fixed and let N be an open set of Y . Suppose u¯ ∈ U is such that f (x, ¯ u) ¯ ∈ N . By continuity of f (·, u) ¯ at x, ¯ there is a neighborhood V of x¯ such that f (x, u) ¯ ∈ N for every x ∈ V . Therefore, for every x ∈ V , we get F (x) ∩ N = ∅.   It is natural to expect that for a given multifunction F : X → P(Y ), there exists a function f : X → Y such that f (x) ∈ F (x) for x ∈ X. The existence of such a function f , called a selector for F , follows from (see [48], Corollary 2.1 of Chap. 2) Zermelo axiom of choice. The most difficult part is to deduce the existence of selectors with prescribed properties. In what follows, we shall present some results dealing with the existence of continuous, measurable, and Lipschitz continuous selectors. The following example shows that continuous set-valued mappings need not have, in general, continuous selectors. Example 2.1.1 Let F be the set-valued mapping defined on the interval (−1, 1) by setting

F (t) =

⎧ ⎨ {(v1 , v2 ) : v1 = cos θ, v2 = t sin θ and ⎩

{(v1 , v2 ) : −1 ≤ v1 ≤ 1, v2 = 0}

≤ θ ≤ 1t + 2π − |t|} for t ∈ (−1, 2) \ {0}, for t = 0 .

1 t

For t = 0 and t ∈ (−1, 1), F (t) is a subset of an ellipse in R2 , whose minor axis shrinks (see Figure 2.2) to zero as t → 0, so that the ellipse collapses to a segment F (0). The subset of the ellipse given by F (t) is obtained by removing a section, from the angle (1/t) − |t| to the angle (1/t). As t gets smaller, the arc length of this hole decreases, while the initial angle increases like (1/t), i.e., it spins around the origin with increasing angular velocity. However, F is continuous at the origin,

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Fig. 2.2 The mapping F

while no selector f : (−1, 0) → R2 or g : (0, 1) → R2 , for example, f (t) = (cos(1/t), t sin(1/t)), can be continuously extended to the whole interval (−1, 1). In fact, the hole in the ellipse would force this selector to rotate around the origin with an angle ρ(t) between (1/t) and (1/t) + 2π − |t| and limt→0 f (t) cannot exist. Then the above defined set-valued mapping does not possess a continuous selector.   We shall show that in some special cases lower semicontinuous multifunctions possess continuous selectors. This follows from the famous Michael continuous selection theorem. We precede it by the following lemmas. Lemma 2.1.2 Let (X, ρ) and (Y, | · |) be a metric and a Banach space, respectively, and let  : X → P(Y ) be a convex-valued and l.s.c. multifunction. Then for every ε > 0 there is a continuous function ϕ : X → Y such that dist(ϕ(x), (x)) ≤ ε for x ∈ X. Proof Let x ∈ X be fixed and select yx ∈ (x) and δx > 0 such that (yx + εK0 ) ∩ (x  ) = ∅ for every x  ∈ Bx , where Bx = B(x, δx ). Since X is paracompact, then there exists a locally finite refinement {Uz }z∈ of {Bz }z∈X . Let {px }x∈ be a partition of unity subordinated to it, and define a function ϕ : X → Y by setting ϕ(x) = z∈ pz (x)yz for x ∈ X. It is clear that ϕ is a continuous function on X. Furthermore, we have x ∈ Uz ⊂ Bz whenever pz (x) > 0. Hence it follows that yz ∈ V ((x), ε). Since this set is convex, every convex combinations of such elements yz s, in particular ϕ(x), belong to this set, too. Therefore, dist(ϕ(x), (x)) ≤ ε for x ∈ X.   Lemma 2.1.3 Let (X, d) and (Y, ρ) be metric spaces, let G : X → P(Y ) be l.s.c., and let g : X → Y be continuous on X. If a real-valued function X  x → ε(x) ∈ R+ is lower semicontinuous on X, then the set-valued mapping  : X → P(Y ) defined by (x) = B(g(x), ε(x)) ∩ G(x) is l.s.c. at every x ∈ X such that (x) = ∅.

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65

Proof Let x¯ ∈ X be such that (x) ¯ = ∅. Select y¯ ∈ (x) ¯ and let η > 0. Assume ε(x) ¯ > ρ(y, ¯ g(x)) ¯ and let σ > 0 be such that ρ(y, ¯ g(x)) ¯ = ε(x) ¯ − σ . There exists σ1 > 0 such that to every x ∈ X with d(x, x) ¯ < σ1 we can associate yx ∈ G(x) ¯ < min(η, (1/3)σ ). Moreover we can select σ2 > 0 such that such that ρ(yx , y) d(x, x) ¯ < σ2 implies ε(x) > ε(x) ¯ − (1/3)σ , and σ3 > 0 such that d(x, x) ¯ < σ3 implies ρ(g(x), ¯ g(x)) < (1/3)σ . Thus ¯ + ρ(y, ¯ g(x))+ ¯ ρ(yx , g(x)) ≤ ρ(yx , y) ρ(g(x), ¯ g(x)) < (1/3)σ + ε(x) ¯ − σ + (1/3)σ = ε(x) ¯ − (1/3)σ < ε(x), whenever d(x, x) ¯ < min{σ1 , σ2 , σ3 }. Then yx ∈ (x) and ρ(yx , y) < η.

 

Now we can prove Michael continuous selection theorem. Theorem 2.1.1 (Michael) Let (X, ρ) and (Y, |·|) be a metric and a Banach space, respectively, and let F : X → P(Y ) be l.s.c. with closed convex values. Then there exists a continuous function f : X → Y such that f (x) ∈ F (x) for x ∈ X. Proof By virtue of Lemma 2.1.2, for ε1 = 1/2 and  = F there exists a continuous function f1 : X → Y such that dist(f1 (x), F (x)) ≤ ε1 for x ∈ X. Let 1 (x) = B(f1 (x), ε1 ) ∩ F (x) for x ∈ X. We have 1 (x) = ∅ for x ∈ X. By Lemma 2.1.3, the multifunction 1 is l.s.c. Then by Lemma 2.1.2, for ε2 = (1/2)2 there exists a continuous function f2 : X → Y such that dist(f2 (x), 1 (x)) ≤ ε2 for x ∈ X. Thus dist(f2 (x), F (x)) ≤ ε2 and dist(f2 (x), B(f1 (x), ε1 )) ≤ ε2 , i.e., f2 (x) − f1 (x) ∈ B(0, ε1 + ε2 ) for x ∈ X. Continuing the above procedure we can deduce that for every εn = (1/2)n , with n = 0, 1, 2, . . ., there exists a continuous function fn : X → Y such that dist(fn (x), F (x)) ≤ εn and fn (x) − fn−1 (x) ∈ B(0, εn−1 + εn ) for x ∈ X. Hence in particular, it follows that supx∈X |fn (x)−fn−1 (x)| ≤ εn−1 +εn for n ≥ 1, which implies that (fn )∞ n=1 is a Cauchy sequence in the Banach space C(X, Y ) of all continuous bounded functions g : X → Y with the supremum norm. Then there exists a continuous function f : X → Y such that supx∈X |fn (x) − f (x)| → 0 as n → ∞. Thus f (x) ∈ F (x) for x ∈ X, because F (x) is a closed subset of Y and dist(fn (x), F (x)) ≤ εn for x ∈ X and n = 1, 2, . . ..   Remark 2.1.1 It can be proved that if the assumptions of Theorem 2.1.1 are satisfied, and (Y, |·|) is separable, then there exists a sequence (fn )∞ n=1 of continuous selectors of F such that F (x) = cl{fn (x) : n ≥ 1} for every x ∈ X.   Remark 2.1.2 There are closed convex-valued u.s.c. multifunctions that do not possess continuous selectors. A simple example is the set-valued mapping F defined by the formula ⎧ ⎨ {−1} for x < 0 F (x) = [−1, 1] for x = 0 ⎩ {+1} for x > 0. with the graph presented in Figure 2.3.

 

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Fig. 2.3 The mapping F

Fig. 2.4 Approximation selector of a set-valued mapping

It can be proved that the above set-valued mapping possesses an approximation continuous selector, i.e., such that for every ε > 0 it belongs (see Figure 2.4) to V (F (x), ε). Now we consider the problem of the existence of more regular selectors of multifunctions. Such selectors are connected with special properties of the “Steiner point map” γ : Conv(Rd ) → Rd defined by " γ (A) =

(d/2) [s(1, A) + s(−1, A)]  d 1 y s(y, A) r(dy)

for d = 1 for d > 1

(2.1.1)

for A ∈ Conv(Rd ), where 1 is the boundary of an open unit ball of Rd and r(·) denotes a surface measure on 1 proportional to the Lebesgue measure and such that r(1 ) = 1. As usual, s(·, A) denotes the support functions of a set A ∈ Conv(Rd ) and Conv(Rd ) denotes a family of all nonempty convex compact subsets of the space Rd . From (2.1.1) and elementary calculations it follows that (i) γ ({x}) = x for every x ∈ Rd . Furthermore, (ii) γ (A + B) = γ (A) + γ (B) and (iii) γ (αA) = αγ (A) for A, B ∈ Conv(Rd ) and α ∈ R. Indeed, for every A, B ∈ Conv(Rd ), one obtains

2.1 Continuity of Multifunctions

67

 γ (A + B) = d

s(y, A + B)y r(dy) = 1



 y s(y, A) r(dy) + d

d 1

y s(y, B) r(dy) = γ (A) + γ (B). 1

Quite similarly, we also get γ (αA) = αγ (A) for α ∈ R and A ∈ Conv(Rd ). Then conditions (ii) and (iii) are also satisfied. We shall show that for every A ∈ Conv(Rd ), one has γ (A) ∈ A. To prove this, let us recall some properties of the group O(Rd ) of all orthogonal linear transformations on Rd . It can be verified that γ (l[A]) = l[γ (A)] for every l ∈ O(Rd ) and A ∈ Conv(Rd ). It is also known that the surface measure r(·) on 1 is invariant under the action of elements in O(Rd ). Lemma 2.1.4 For every A ∈ Conv(Rd ) one has γ (A) ∈ A. Proof Suppose there is A ∈ Conv(Rd ) such that γ (A) ∈ A. Define C = A − γ! (A). Then 0 ∈ C, and by (i)–(iii), we get γ (C) = 0. Let 0 = cˆ be such that c − c, ˆ xˆ > 0 −1 , and ·, · denotes the inner product in Rd . But for every c ∈ C, where x ˆ = c ˆ c ˆ ! ! ! ! ! c, xˆ = cˆ + (c − c), ˆ xˆ = c, ˆ xˆ + c − c, ˆ xˆ and c, ˆ xˆ = c. ˆ Then for every ! c ∈ C, one has c ˆ ≤ c, xˆ . Let l : Rd → Rd be the linear transformation defined by l(x) ˆ = xˆ and l(x) = −x for x ∈ Rd orthogonal to x. ˆ It can be verified that l belongs to the group O(Rd ) of orthogonal linear transformations on Rd and l 2 = I , the identity map. So l = l ∗ . Let D = C + l(C). Then l(D) = D and so γ (D) = 0. ! In addition, for every d ∈ D we have d, xˆ ≥ 2c ˆ > 0 and so 0 ∈ D. Now let ! ! 10 = {y ∈ 1 : y, xˆ = 0}, 1+ = {y ∈ 1 : y, xˆ > 0} and ! 1− = {y ∈ 1 : y, xˆ < 0}. Then 1 = 10 ∪ 1+ ∪ 1− and these three sets 10 , 1+ , 1− are disjoint. Also r(10 ) = 0. So we have 

 γ (D) = d

1+

y s(y, D) r(dy) + d

1−

y s(y, D) r(dy) =

 y[s(y, D) − s(−y, D)] r(dy).

d 1

Let y ∈ 1+ and e ∈ D be such that s(−y, D) = −y, e . Then s(y, D) − s(−y, D) = s(y, l(D)) − s(−y, D) = s(l(y), D) − s(−y, D) ≥ l(y), e + y, e = (l + I )(y), e .

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! ! ! But (l + I )(y) = 2 y, xˆ x. ˆ Then s(y, D) − s(−y, D) ≥ 2 y, xˆ · x, ˆ e > 0 since ! y ∈ 1+ and x, ˆ e > 0. Therefore, ! γ (D), xˆ = d



! [s(y, D) − s(−y, D)] · y, xˆ r(dy) > 0, 1

which contradicts to γ (D) = 0. Then γ (A) ∈ A for every A ∈ Conv(Rd ).

 

Corollary 2.1.1 There is K(d) > 0 such that for every A, B ∈ Conv(Rd ), one has |γ (A) − γ (B)| ≤ K(d) · h(A, B). Proof Let us observe (see Corollary 1.3.2 of Chapter 1) that for A, B ∈ Conv(Rd ) we have h(A, B) = max{|s(x, A) − s(x, B)| : |x| = 1}. Then |γ (A) − γ (B)| ≤ d 1 y |s(y, A) − s(y, B)| r(dy) ≤ K(d) · h(A, B), for every K(d) ≥ d.   Remark 2.1.3 In the above inequality we can compute the optimal Lipschitz constant K(d) > 0. It is equal to d!!/(d − 1)!! if d is odd, and K(d) = d!!/[π(d − 1)!!] if d is even.   We can prove now the existence of Lipschitz continuous selectors for Lipschitz continuous multifunctions defined on a given metric space (X, ρ) with values in the space Conv(Rd ). Let us recall that a multifunction F : X → Conv(Rd ) is Lipschitz continuous, if there exists an number L > 0 such that h(F (x), F (y)) ≤ Lρ(x, y) for every x, y ∈ X. Theorem 2.1.2 If (X, ρ) is a metric space and F : X → Conv(Rd ) is Lipschitz continuous, then F admits a Lipschitz continuous selector. Proof Let h(F (x1 ), F (x2 )) ≤ Lρ(x1 , x2 ) for some L > 0 and every x1 , x2 ∈ X. Put f (x) = γ (F (x)) for x ∈ X. By Corollary 2.1.1, we get |f (x1 ) − f (x2 )| = |γ (F (x1 ))−γ (F (x2 ))| ≤ K(d)·h(F (x1 ), F (x2 )) ≤ K(d)·L ρ(x1 , x2 ), where K(d) is as in Remark 2.1.3. By Lemma 2.1.4, for every x ∈ X we have f (x) ∈ F (x).   Remark 2.1.4 Theorem 2.1.2 cannot be extended to multifunctions with values in an infinite-dimensional Banach space (Y, | · |). It can be proved (see [87]) that if a Lipschitz continuous multifunction F : X → Conv(Y ) admits a Lipschitz continuous selector, then Y is finite dimensional.   Focusing in particular case, on a multifunction F : I → Conv(Rd ) whose domain is an interval I ⊂ R, we may define the notion of the Hukuhara derivative (DH F )(t0 ) at point t0 ∈ I if there exists a set C ∈ Conv(X) such that limh→0+ (1/ h)[F (t0 + h) − F (t0 )] = limh→0 (1/ h)[F (t0 ) − F (t0 − h)] = C, where for given sets A, B ∈ Conv(Rd ) by A − B the Hukuhara difference (see Section 1.3 of Chapter 1) between A and B is denoted. The set C is then called the Hukuhara derivative of F at t0 ∈ I and denoted by (DH F )(t0 ). It is important also to refer that if we restrict ourselves to single valued mappings, then the previous notation reduces to their classical counterparts, i.e., to the ordinary derivative of vector-valued function.

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69

2.2 Measurability of Multifunctions Let (T , F) be a measurable space and (X, d) a separable metric space. A set-valued mapping F : T → P(X) is said to be measurable (weakly measurable) if for every closed (open) set E ⊂ X, we have F − (E) = : {t ∈ T : F (t) ∩ E = ∅} ∈ F. It is clear that if F is measurable, then it is weakly measurable. The converse statement is true if F (t) ∈ Comp(X). It is not true in general. A multifunction F : T → P(X) is said to be graph measurable if Graph(F ) ∈ F ⊗ β(X), where β(X) is the Borel σ -algebra on X. It can be verified (see [27], Prop.1.7 of Chap. 2) that a measurable multifunction is graph measurable. Remark 2.2.1 Let (T , F) be a measurable space and X a separable metric space. For a given F : T → P(X), let us denote by F the set-valued mapping F : T → P(X) with F (t) = cl[F (t)] for every t ∈ T . Similarly, if X is a separable normed space, then co F denotes the set-valued mapping co F : T → P(X) with (co F )(t) = co[F (t)] for every t ∈ T . It is clear that F and co F are measurable whenever F is measurable, because F (t) ⊂ F (t) and F (t) ⊂ coF (t) for every t ∈ T.   Remark 2.2.2 If (T , F) is a measurable space, (X, | · |) is a separable Banach space and F : T → Cl(X) is measurable, then the function T  t → s(x ∗ , F (t)) ∈ R is measurable for every x ∗ ∈ X∗ . If F : T → Cl(X) is convex-valued, then F is measurable if and only if s(x ∗ , F (·)) is measurable for every x ∗ ∈ X∗ .   We have the following result dealing with measurable multifunctions. Theorem 2.2.1 Let (X, ρ) be a separable metric space and (T , F) a measurable space. Then a multifunction F : T → P(X) is weakly measurable if and only if the function T  t → dist(x, F (t)) ∈ R+ is measurable for each x ∈ X. Proof Let us observe that F is weakly measurable if and only if F − (B(x, ε)) ∈ F for every x ∈ X and ε > 0. On the other hand, a function T  t → dist(x, F (t)) ∈ R+ is for a fixed x ∈ X measurable if and only if {t ∈ T : dist(x, F (t)) < ε} ∈ F for every ε > 0. But F − (B(x, ε)) = {t ∈ T : F (t) ∩ B(x, ε) = ∅} = {t ∈ T : dist(x, F (t)) < ε}. Then F − (B(x, ε)) ∈ F.   We shall now consider the problem of the existence of measurable selectors of measurable multifunctions. Let us recall that for a given measurable multifunction F : T → P(X), by a measurable selector for F we mean a measurable function f : T → X such that f (t) ∈ F (t) for t ∈ T . The existence of measurable selectors follows from the following theorem. Theorem 2.2.2 (Kuratowski and Ryll-Nardzewski) Let (X, ρ) be a Polish space and (T , F) a measurable space. If F : T → Cl(X) is measurable, then F admits a measurable selector. Proof Let {x1 , x2 , . . .} be a countable and dense subset in X, and let Bn (i) = {x ∈ X : ρ(x, xi ) ≤ 1/n} for i, n ≥ 1. Without any loss of the generality we

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may assume that diam(X) < 1, where diam(X) = sup{ρ(x, y) : x, y ∈ X}. We will construct a sequence (fn )∞ n=1 of measurable functions fn : T → X such that (i) dist(fn (t), F (t)) ≤ εn and (ii) ρ(fn (t), fn−1 (t)) ≤ εn−1 for n ≥ 0 and t ∈ T , where εn = (1/2)n for n = 0, 1, 2, . . .. Let f0 (t) = x1 for t ∈ T . Then dist(f0 (t), F (t)) < 1. Suppose f0 , . . . , fn−1 have been constructed, and let Ank = {t ∈ T : dist(fk (t), F (t)) < εn } and Ckn = {t ∈ T : ρ(xk , fn−1 (t)) < εn−1 }. n n n Put Dk = Ak ∩ Ck . We claim that T = k≥1 Dkn for n ≥ 1. Fix t ∈ T . By the inductive hypothesis, we can find z ∈ F (t) such that ρ(fn−1 (t), z) < εn−1 . On the other hand, there is k ≥ 1 such that ρ(xk , z) < εn and ρ(xk , z)+ ρ(z, fn−1 (t)) < εn + εn−1 < 2εn−2 = εn−1 . Therefore, t ∈ Dkn and T ⊂ k≥1 Dkn . By virtue of Theorem 2.2.1, and continuity of the function dist(·, F (t)) for fixed t ∈ T we obtain that Ank ∈ F. The inductive hypothesis gives that Ckn ∈ F. Then Dkn ∈ F.  n Now define fn : T → X by setting fn (t) = xk for t ∈ Dkn \ k−1 i=1 Di . Clearly fn is ∞ measurable. Moreover, by (ii) we see that (fn (t))n=1 is a Cauchy sequence in X for every fixed t ∈ T . Then there exists a function f : T → X such that fn (t) → f (t) for every t ∈ T as n → ∞. We also have dist(f (t), F (t)) = 0 for every t ∈ T . Hence it follows that f is measurable and such that f (t) ∈ F (t) for every t ∈ T .   In what follows, we shall consider “complete” measurable spaces defined in the following way. For a given measurable space (T , F) and every probability  measure = μ on F, we denote by Fμ the μ-completion of F and define F μ Fμ . The space (T , F) is said to be complete if F = F. Remark 2.2.3 It can be proved (see [27], Theorem 2.14 of Chap. 2) that for a given complete measurable space (T , F), a graph measurable multifunction F : T → P(Rn ) admits a measurable selector.   The following important result follows from Theorem 2.2.2. Theorem 2.2.3 Let (X, ρ) be a Polish space, (T , F) a measurable space and let F : T → Cl(X). The following conditions are equivalent (a) F is weakly measurable, (b) there exists a sequence (fn )∞ n=1 of measurable selectors of F such that F (t) = cl{f1 (t), f2 (t), . . .} for every t ∈ T , (c) if there exists a complete σ -finite measure on F, then F is measurable if and only if it is graph measurable. Proof Let F be measurable and (xn )∞ n=1 be a dense sequence in X. For every n, k ≥ 1 we define  F (t) ∩ B(xn , εk ) if t ∈ F − (B(xn , εk )) Fn,k (t) = F (t) otherwise , where εk = (1/2)k and F − (B(xn , εk )) = {t ∈ T : F (t)∩B(xn , εk ) = ∅}. Note that F − (B(xn , εk )) ∈ F and that the set-valued function T  t → F (t)∩B(xn , εk ) ⊂ X

2.2 Measurability of Multifunctions

71

is measurable. So Fn,k is measurable, which implies that F n,k is also measurable. Therefore, by the Kuratowski and Ryll-Nardzewski measurable selection theorem, there exist measurable functions fn,k : T → X such that fn,k (t) ∈ F n,k (t) for every t ∈ T . We shall show that F (t) = cl{fn,k (t) : n, k ≥ 1} for t ∈ T . Indeed, fix t ∈ T and ε > 0 and let x ∈ F (t). Let k ≥ 1, and n ≥ 1 be such that εk−1 ≤ ε, and x ∈ B(xn , εk ). Then t ∈ F − (B(xn , εk )) and fn,k (t) ∈ B(xn , εk ). So ρ(fn,k (t), x) ≤ ρ(fn,k (t), xn )+ρ(xn , x) ≤ ε, which proves that F (t) = cl{fn,k (t) : n, k ≥ 1}. Thus (a) ⇒(b). Assume that (b) is satisfied. Then for every open set U ⊂ X we have F − (U ) = {t ∈ T : F (t) ∩ U = ∅} =



{t ∈ T : fn (t) ∈ U } ∈ F.

n≥1

Therefore, F is weakly measurable. Thus (b) ⇒(a). The proof of the equivalency (c)⇔(a) can be found in [26].   Theorem 2.2.4 Assume that (X, ρ) is a Polish space, (T , F) a measurable space, and (Y, d) a metric space. Suppose f : T × X → Y is a function measurable in t ∈ T and continuous in x ∈ X, and let  : T → Comp(X) be a measurable multifunction and g : T → Y a measurable function such that g(t) ∈ f (t, (t)) for t ∈ T . Then there exists a measurable function γ : T → X such that γ (t) ∈ (t) and g(t) = f (t, γ (t)) for t ∈ T . Proof Let us observe that the set-valued function F : T → P(X) defined by F (t) = {x ∈ X : f (t, x) ∈ U } for t ∈ T is measurable for every open set U ⊂ Y . Indeed, let B be a closed subset of X and let A be a countable dense subset of B. We have F − (B) = {t ∈ T : F (t) ∩ B = ∅} = {t ∈ T : f (t, x) ∈ U for some x ∈ B} = {t ∈ T : f (t, a) ∈ U for some a ∈ A} =



{t ∈ T : f (t, a) ∈ U }.

a∈A

Therefore, F − (B) ∈ F, because we have {t ∈ T : f (t, a) ∈ U } ∈ F for every fixed a ∈ A. Define multifunctions H (t) = (t) ∩ {x ∈ X : d(f (t, x), g(t)) = 0} for t ∈ T and Fn (t) = {x ∈ X : d(f (t, x), g(t)) < 1/n} for t ∈ T and n ≥ 1. For every n = 1, 2, . . ., a multifunction Fn is measurable, and therefore also weakly measurable. Hence it follows that its closure F¯n is weakly measurable, because − Fn− (B)= F¯n (B) for every open set B ⊂ X. Clearly {x ∈ X : d(f (t, x), g(t)) = ∞ 0} = n=1 F¯n (t) for t ∈ T , because F¯n (t) ⊂ {x ∈ X : d(f (t, x), g(t)) ≤ 1/n} for t ∈ T and n ≥ 1. Hence it follows the defined above multifunction H #that $ ∞ ¯ can be also defined by H (t) = (t) ∩ n=1 Fn (t) for t ∈ T which implies that H is measurable. Therefore, by the Kuratowski and Ryll-Nardzewski measurable selection theorem, there exists a measurable selector γ for H, that in particular is a selector for  satisfying d(f (t, γ (t)), g(t)) = 0 for t ∈ T .  

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Corollary 2.2.1 Let (X, ρ) be a Polish space, and (T , F) a measurable space. If  : T → Comp(X) and g : T → X are measurable, then there exists a measurable selector γ for  such that dist(g(t), (t)) = ρ(g(t), γ (t)) for t ∈ T .   We shall consider now the existence of the Carathèodory selectors of measurable multifunctions depending on two variables. More precisely, assume that (T , F) is a measurable space, (X, ρ) is a Polish space, and (Y, | · |) is a separable Banach space. Consider the set-valued mapping F : T × X → Cl(Y ), which is assumed to be measurable, i.e., such that for every closed set A ⊂ Y we have F − (A) = {(t, x) ∈ T × X : F (t, x) ∩ A = ∅} ∈ F ⊗ β(X). We are interested in the existence of a function f : T × X → Y , a selector of F , such that f (·, x) is measurable for fixed x ∈ X, and f (t, ·) is continuous for fixed t ∈ T . Such selectors of F are said to be of Carathèodory type or simply Carathèodory selectors for F . Theorem 2.2.5 Let (T , F) be a complete measurable space, (X, ρ) a Polish space, and (Y, | · |) a separable Banach space. Assume that F : T × X → Cl(Y ) is a measurable convex-valued multifunction. If furthermore F (t, ·) is l.s.c. for fixed t ∈ T , then F admits a Carathèodory selector. Proof Let (yn )∞ n=1 be a dense sequence of Y . For t ∈ T , n ≥ 1, and ε > 0 define Gεn (t) = {x ∈ X : yn ∈ V (F (t, x), ε)}. By the lower semicontinuity of F (t, ·), a set Gεn (t) is open for every t ∈ T , ε > 0 and n ≥ 1. Also, the family {Gεn (t) : n ≥ 1} is an open covering of X. Moreover, Graph(Gεn ) = {(t, x) ∈ T × X : dist(yn , F (t, x)) < ε} ∈ F ⊗ β(X), (1/2)m and Gεn,m (t) = {x ∈ Gεn (t) : dist(x, X \ because F is measurable. Let εm = ε ε ε Gn ) ≥ εm }, and Un (t) = Gn (t) \ 1≤k 0 and n ≥ 1. Furthermore, ε (t, x) = 1. Let f ε (t, x) = ε p n≥1 n n≥1 pn (t, x) · yn . It is clear that f is a Carathèodory function. By the convexity of F (t, x) for every (t, x) ∈ T × X we get f ε (t, x) ∈ V (F (t, x), ε) for (t, x) ∈ T × X and every ε > 0. Let εn = (1/2)n for n = 1, 2, . . .. We define now a sequence (fn )∞ n=1 of Carathèodory functions fn : T × X → Y such that fn (t, x) ∈ V (F (t, x), εn ) and |fn (t, x) − fn−1 | < εn−1 for (t, x) ∈ T ×X and n ≥ 2. Let f1 = f ε1 , and put F2 (t, x) = F (t, x)∩B(f1 (t, x), ε1 ) for (t, x) ∈ T × X. By virtue of Lemma 2.1.3, the multifunction F2 (t, ·) is l.s.c. for fixed t ∈ T . It is easy to see that F2 is measurable. Consequently, its closure F 2 is

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73

measurable and F 2 (t, ·) is l.s.c. for fixed t ∈ T . From this and the first part of the proof, it follows that for ε = ε2 , there exists a Carathèodory function f2 such that f2 (t, x) ∈ V (F2 (t, x), ε2 ) for (t, x) ∈ T × X. It is clear that |f2 (t, x) − f1 (t, x)| < ε1 for (t, x) ∈ T × X. By the inductive procedure we can define a sequence (fn )∞ n=1 of Carathèodory functions fn : T ×X → Y such that fn (t, x) ∈ V (F (t, x), εn ), and |fn (t, x) − fn−1 (t, x)| < εn−1 for (t, x) ∈ T × X. Hence it follows that there exists a Carathèodory function f : T × X → Y such that fn (t, x) → f (t, x) as n → ∞ for (t, x) ∈ T × X. By the closedness of F (t, x) it implies that f (t, x) ∈ F (t, x) for (t, x) ∈ T × X.   Remark 2.2.4 It can be proved that if T is a locally compact metric space furnished with a Radon measure μ, X is a Polish space, Y is a separable reflexive Banach space, and F : T × X → Cl(Y ) is such as in Theorem 2.2.5, then there exists a sequence (fm )∞ m=1 of Carathèodory selectors fm : T × X → Y of F such that F (t, x) = cl{fm (t, x) : m ≥ 1} for every (t, x) ∈ T × X.  

2.3 Subtrajectory Integrals Given a σ -finite measure space (T , F, μ) and a separable Banach space (X, | · |), by M(T , X) we shall denote the family of all measurable functions f : T → X. Let p ≥ 1, and let Lp (T , F, μ, X) denote the space ofall Bochner integrable functions (equivalence classes of) f : T → X such that T |f (t)|p μ(dt) < ∞. The precise definition of this space and its properties are presented in Section 3.1 of Chapter 3. In this section, for simplicity, the space Lp (T , F, μ, X) will be denoted by Lp (T , X). Given a sub-σ -algebra G of F a set K ⊂ M(T , X) is said to be Gdecomposable if for every f, g ∈ K and every A ∈ G one has 1A f + 1A∼ g ∈ K, where A∼ = T \ A. In particular, an F-decomposable set K ⊂ M(T , X) will be simply called decomposable. Similarly as convex hulls of subsets of linear vector spaces, for the given set K ⊂ M(T , X) its decomposable hull can be defined. It is denoted by dec K, and defined to be the smallest decomposable subset of M(T , X) containing K. The closed decomposable hull dec K of the set K ⊂ Lp (T , X) is defined in a similar way, i.e., it is the smallest closed decomposable subset of the space Lp (T , X) containing K. In a similar way, for a given sub-σ -algebra G of F and a set K ⊂ M(T , X), a G-decomposable hull decG K of K can be defined, i.e., it is the smallest G-decomposable set of M(T , X) containing K. The closed G-decomposable hull decG K of a set K ⊂ Lp (T , X) is defined in a similar way. It is clear that decG K ⊂ dec K and decG K ⊂ dec K. Remark 2.3.1 Similarly to the theory of convex hulls of subsets of linear spaces (see Lemma 1.2.7 of Chapter 1), it can be proved that for a given  set K ⊂ M(T , X) the set dec K consists of all decomposable combinations m k=1 1Ak xk , with x1 , . . . , xm ∈ K, and the finite F-measurable partition {A1 , A2 , . . . , Am } of T.  

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For a given multifunction F : T → Cl(X) its subtrajectory integrals S(F ) is defined by setting S(F ) = {f ∈ Lp (T , X) : f (t) ∈ F (t) a.e.}. If G is a sub-σ algebra of F, then SG (F ) denotes a subset of S(F ) containing all its G-measurable elements. In particular, we have S(F ) = SF (F ). If needed, the set S(F ) is denoted by S p (F ) to identify it as a subset of the space Lp (T , X). For simplicity we use this notation only in some special cases. It can be verified that S(F ) is a closed decomposable subset of Lp (T , X). In what follows, by A(T , X) we shall denote the family of all measurable multifunctions F : T → Cl(X) such that S(F ) = ∅. It can be verified that F ∈ A(T , X) if and only if there exists k ∈ Lp (T , R+ ) such that dist(0, F (t)) ≤ k(t) for a.e. t ∈ T . If h(F (t), {0}) ≤ k(t) for a.e. t ∈ T , then F is said to be p-integrably bounded. In particular, if k is a constant function, then a multifunction F is said to be bounded. We have the following results. Theorem 2.3.1 If F ∈ A(T , X), then there exists a sequence (fn )∞ n=1 of functions fn ∈ S(F ) such that F (t) = cl{f1 (t), f2 (t), . . .} for a.e. t ∈ T . Proof By virtue of Theorem 2.2.3, there exists a sequence (gn )∞ n=1 of measurable functions gn : T → X such that F (t) = cl{g1 (t), g2 (t), . . .} for t ∈ T . Taking a countable measurable partition {A1 , A2 , . . .} of T with μ(Ak ) < ∞, and a function f ∈ Lp (T , X) such that f (t) ∈ F (t) for t ∈ T , we define Bj,m,k = {t ∈ T : m − 1 ≤ |gj (t)| < m} ∩ Ak and fj,m,k = 1Bj,m,k gj + 1T \Bj,m,k f for j, m, k ≥ 1. It  is easy to see that fj,m,k ∈ S(F ) and F (t) = {fj,m,k (t) : j, m, k ≥ 1} for t ∈ T .  Corollary 2.3.1 Let (T , F, μ) be a σ -finite measure space and G be a sub-σ algebra of F. If F : T → Cl(X) and G : T → Cl(X) are p-integrably bounded, and G- and F-measurable, respectively, such that SG (F ) ⊂ SF (G), then F (t) ⊂ G(t) for a.e. t ∈ T . In particular, if F and G are F-measurable, then SF (F ) = SF (G) if and only if F (t) = G(t) for a.e. t ∈ T . Proof We have SG (F ) = ∅. Then by Theorem 2.3.1, there exists a sequence (fn )∞ n=1 ⊂ SG (F ) such that F (t) = cl{fn (t) : n ≥ 1} for a.e. t ∈ T . But, SG (F ) ⊂ SF (G) and G ⊂ F. Then (fn )∞ n=1 ⊂ SF (G), which implies that cl{fn (t) : n ≥ 1} ⊂ G(t) for a.e. t ∈ T . Therefore, F (t) ⊂ G(t) for a.e. t ∈ T .   Theorem 2.3.2 Let F ∈ A(T , X) and let (fn )∞ n=1 be a sequence of S(F ) such that F (t) = cl{f1 (t), f2 (t), . . .} for a.e. t ∈ T . Then for every f ∈ S p (F ) with p ≥ 1, and ε > 0 there exists a finite F-measurable mpartition {A1 , . . . , Am } of T and a ∞ such that | f − family (fnk )m ⊂ (f ) n k=1 1Ak fnk | < ε, where | · | is the n=1 k=1 norm of the space Lp (T , X) with p ≥ 1. Proof Assume f (t) ∈ F (t) for every t ∈ T , and let ρ ∈ L1 (T , R) be strictly positive and such that T ρ dμ < 2εp /3. Then there exists a countable measurable ∞ partition {B1 , B2 , . . .} of T and a subsequence (fnk )∞ k=1 of (fn )n=1 such that m such that |f (t) − fnk (t)| < ρ(t) for t ∈ Bk and k≥ 1. Take  an integer  ∞ ∞ p dμ < (ε/2)p /3 and p dμ < (ε/2)p /3, |f (t)| |f (t)| k=m+1 Bk nk k=m+1 Bk and ∞define a finite measurable partition {A1 , . . . , Am } as follows: A1 = B1 ∪ ( i=m+1 Bi ) and Aj = Bj for 2 ≤ j ≤ m. Then we have

2.3 Subtrajectory Integrals

|f−

m  k=1

75

 1Ak fnk | p ≤

ρ dμ+ 2p−1 T

∞  

(|f (t)|p +|fnk (t)|p )dμ < εp .

k=m+1 Bk

  Corollary 2.3.2 For every F ∈ A(T , X), one has S(F ) = dec{fn : n ≥ 1}, where fn ∈ S(F ) for n ≥ 1 are such that F (t) = cl{fn (t) : n ≥ 1} for a.e. t ∈ T . Proof By the properties of S(F ) we have dec{fn : n ≥ 1} ⊂ S(F ). On the other hand, by virtue of Theorem 2.3.2, for every f ∈ S(F ) and ε > 0 there exist an N F-measurable partition (Ak )N k=1 of T and a family (fnk )k=1 ⊂ {fn : n ≥ 1} such N that | f − k=1 1Ak fnk | ≤ ε, which implies that f ∈ dec{fn : n ≥ 1}, because  by Remark 2.3.1 one has N k=1 1Ak fnk ∈ dec{fn : n ≥ 1}. Thus, S(F ) = dec{fn : n ≥ 1}.   We prove now some representation theorem for decomposable sets, which ties them with subtrajectories of multifunctions belonging to the family A(T , X). Theorem 2.3.3 Let M be a nonempty closed subset of Lp (T , X) with p ≥ 1. Then there exists an F ∈ A(T , X) such that M = S(F ) if and only if M is decomposable. Proof Let us observe that for every F ∈ A(T , X), the set S(F ) is a closed decomposable subset of the space Lp (T , Rd ). If M ⊂ Lp (T , X) is such that there exists F ∈ A(T , X) such that M = S(F ), then M it is closed and decomposable. To prove the converse, assume that M is a nonempty closed decomposable subset of Lp (T , X). Let us observe that a multifunction G defined by G(t) = X for every t ∈ T belongs to A(T , X). Therefore, by virtue of Theorem 2.3.1, there exists p a sequence (fi )∞ i=1 of L (T , X) such that G(t) = cl{(fi (t) : i ≥ 1} for every t ∈ T . Let αi = inf{ | fi − g | : g ∈ M} for i ≥ 1, and choose a sequence {gij : j ≥ 1} ⊂ M such that | fi − gij | → αi as j → ∞. Define F ∈ A(T , X) by setting F (t) = cl{gij (t) : i, j ≥ 1}. We shall prove that M = S(F ). By Theorem 2.3.2, for each f ∈ S(F ) and ε > 0 we can select a finite measurable partition  {A1 , . . . , Am } of T and{hm1 , . . . , hm } ⊂ {gij (t) : i, j ≥ 1} such that |f − m k=1 1Ak hk | < ε. Since k=1 1Ak hk ∈ M, and M is closed, this implies that f ∈ M. Therefore, S(F ) ⊂ M. Now suppose that S(F ) = M. Then there exist f ∈ M, A ∈ F with μ(A) > 0 and δ > 0 such that infi,j ≥1 |f (t) − gij (t)| ≥ δ, for t ∈ A. Take an integer i, fixed in the rest of the proof, and such that the set B = A ∩ {t ∈ T : |f (t) − fi (t)| < δ/3} has a positive measure, and let gj = 1B f + 1T \B gij , for j ≥ 1. Since gj ∈ M for j ≥ 1 and |fi (t) − gij (t)| ≥ |f (t) − gij (t)| − |f (t) − fi (t)| > 2δ/3, then | fi − gij | p − αi ≥ | fi − gij | p − | fi − gj | p =  B

% & |fi (t) − gij (t)|p − |fi (t) − f (t)|p dμ ≥ [(2δ/3)p − (δ/3)p ] · μ(B) > 0

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for j ≥ 1. If j tends to infinity, we get limj →∞ | fi − gij | > αi , a contradiction. Then M = S(F ).   We shall now consider the problem of estimations of integrals of some func be tionals defined on subtrajectory of a given multifunction. Let φ :  × S → R = [−∞, +∞]. Let F ⊗ β(X)-measurable function and F ∈ A(T , X), where R  by setting, Tφ (f ) = us define a functional Tφ : S(F ) → R T φ(t, f (t))μ(dt) for every f ∈ S(F ). We shall prove that by appropriate assumptions, we have  inf{Tφ (f ) : f ∈ S(F )} = T inf{φ(t, x) : x ∈ F (t)}μ(dt). We begin with the following lemma. be F ⊗ β(X)-measurable. Lemma 2.3.1 Let F ∈ A(T , X) and φ : T × S → R Assume either that (i) φ(t, x) is u.s.c. in x for every fixed t ∈ T or that (ii) (T , F, μ) is complete and φ(t, x) is l.s.c. in x for every fixed t ∈ T . Then the function T  is measurable. t → inf{φ(t, x) : x ∈ F (t)} ⊂ R Proof Let ξ(t) = inf{φ(t, x) : x ∈ F (t)} and assume that (i) is satisfied. By Theorem 2.3.1 there exists a sequence (fn )∞ n=1 of measurable selectors of F such that F (t) = cl({f1 (t), f2 (t), . . .}) for t ∈ T . Then we have ξ(t) = infn≥1 φ(t, fn (t)) for t ∈ T , which implies that ξ is measurable. Let (ii) be satisfied and let H : T → P(X × R) be defined by H (t) = {(x, α) ∈ X × R : x ∈ F (t), φ(t, x) ≤ α} for t ∈ T . Then H (t) is closed in X × R for every t ∈ T and Graph(H ) = [Graph(F ) ∩ R] ∩ {(t, x, α) : (t, x) − α ≤ 0} belongs to F ⊗ β(X) ⊗ β(R) = F ⊗ β(X × R). Therefore, by virtue of Theorem 2.3.1 there exists a sequence (gn , ξn )∞ n=1 of measurable functions gn : T → X, and ξn : T → R such that H (t) = cl({(g1 , ξ1 )(t), (g2 , ξ2 )(t), . . .}) for t ∈ Dom(H ). Hence we have ξ(t) = infn≥1 ξn (t) for t ∈ Dom(H ) and ξ(t) = ∞ for t ∈ T \ Dom(H ). This shows that ξ is measurable.   be F ⊗ β(X)-measurable. Theorem 2.3.4 Let F ∈ A(T , X) and φ : T × X → R Assume either that (i) φ(t, x) is u.s.c. in x for every fixed t ∈ T or that (ii) (T , F, μ) is complete and φ(t, x) is l.s.c in x for every fixed t ∈ T . If the integral functional Tφ is defined for all  f ∈ S(F ) and Tφ (f0 ) < ∞ for some f0 ∈ S(F ), then inf{Tφ (f ) : f ∈ S(F )} = T inf{φ(t, x) : x ∈ F (t)}dμ. Proof Let ξ(t) = inf{φ(t, x) : x ∈ F (t)}. By virtue of Lemma 2.3.1, ξ is measurable and ξ(t) ≤ φ(t, f (t)) a.e. for  every f ∈ S(F ). Taking f = f0 we can see that the integral of ξ exists and T ξ dμ ≤ inf{Tφ (f ) : f ∈ S(F )}. If Tφ (f0 ) = −∞, then the proof is complete. Thus assume Tφ (f0) is finite, so that the is in L1 (T , R). Let β > function T  t → φ(t, f0 (t)) ∈ R T ξ dμ be given. We shall show that Tφ (f ) < β for some f ∈ S(F ). Take a sequence (An )∞ n=1 ⊂ F of sets An ∈ F of the finite measure such that An ↑ T , and a strictly positive function ρ ∈ L1 (T , R). For n ≥ 1 define Bn = An ∩ {t ∈ T : φ(t, f0 (t)) ≥ −n} and let ⎧ ⎨ ξ(t) + ρ(t)/n if t ∈ Bn and ξ(t) ≥ −n, ξn (t) = −n + ρ(t)/n if t ∈ Bn and ξ(t) < −n ⎩ φ(t, f0 (t)) + ρ(t)/n if t ∈ T \ Bn .

2.3 Subtrajectory Integrals

77

1  It is easy to see that ξn ∈ L (T , R) for n ≥ 1 and ξn (t) ↓ ξ(t) a.e., so that T ξn0 dμ < β for some n0 . Setting ζ = ξn0 we have T ζ dμ < β and ξ(t) < ζ (t) a.e. We claim that there exists a measurable function g : T → X satisfying g(t) ∈ F (t) a.e. and φ(t, g(t)) ≤ ζ (t) a.e. For case (i), take a sequence (gi )∞ i=1 of measurable functions such that F (t) = cl({g1 (t), g2 (t), . . .} for all t ∈ T . Since infi≥1 φ(t, gi (t)) = ξ(t) a.e., then there exists a measurable function g satisfying the conditions desired above. For case (ii), define F1 (t) = F (t)∩{x ∈ S : φ(t, x) ≤ ζ (t)} for t ∈ T . Since F1 (t) is closed for every t ∈ T and Graph(F1 ) ∈ F ⊗ β(X), it follows by Theorems 2.2.3 and 2.2.2, that F1 has a measurable selector on Dom(F1 ) ∈ F. Thus the desired g is obtained from the condition μ(T \Dom(F1 )) = 0. Using the function g defined above, we define Cn = An ∩ {t ∈ T : |g(t)| ≤ n} and fn = 1Cn g + 1T \Cn f0 for n ≥ 1. Then fn ∈ S(F ) for n ≥ 1 and





Tφ (fn ) =

φ(t, g(t))dμ + Cn

Since

 T

T \Cn





φ(t, f0 (t))dμ ≤

ζ dμ + T

T \Cn

[φ(t, f0 (t)) − ζ ]dμ.

ζ dμ < β and Cn ↑ T , we obtain Tφ (fn ) < β.

 

is F ⊗ β(X)-measurable, Remark 2.3.2 If F ∈ A(T , X) and φ : T × X → R satisfies the condition (i) or (ii) of Theorem 2.3.4, Tφ is defined for all f ∈ S(F ) and  Tφ (f0 ) > −∞ for some f0 ∈ S(F ), then sup{Tφ (f ) : f ∈ S(F )} =   T sup{φ(t, x) : x ∈ F (t)}dμ. We shall now present some properties of subtrajectory integrals. Theorem 2.3.5 Let (T , F, μ) be a σ -finite measure space and let (X, | · |) be a separable Banach space having the Radon–Nikodym property. If F ∈ A(T , X), then co S(F ) = S(co F ). Proof We have co S(F ) ⊂ S(co F ). Assume that there exists f ∈ S(co F ) such that f ∈ co S(F ). By the separation theorem (see Theorem 1.2.2 of Chapter 1), for p = 1, we can find h ∈ L∞ (T , X) such that sup{ h, g : g ∈ co S(F )} < h, f , where ·, · denotes the duality bracket (see Section 1.2 of Chapter 1). By Remark 2.3.2, and the definition of the duality bracket one gets  sup{ h, g : g ∈ co S(F )} = sup

 h(t), u(t) μ(dt) : u ∈ co S(F ) =

T







h(t), u(t) μ(dt) : u ∈ S(F ) =

sup T

sup{ h(t), u(t) : u ∈ F (t)}μ(dt) = T



 sup{ h(t), u(t) : u ∈ co F (t)}μ(dt) = T

s(h(t), co F (t))μ(dt). T

78

2 Multifunctions

  Then T s(h(t), co F (t))dμ < T h(t), f (t) dμ.  On the other hand, f (t) ∈ co F (t) a.e. and therefore, T h(t), f (t) dμ ≤ T s(h(t), co F (t))dμ, a contradiction. Then co S(F ) = S(coF ). In a similar way, for p > 1 the above result can be verified.   Corollary 2.3.3 Let (T , F, μ) be a σ -finite measure space and (X, | · |) be a separable Banach space with the Radon–Nikodym property. Then a multifunction F ∈ A(T , X) is convex-valued if and only if S(F ) is a convex subset of the space Lp (T , F, μ, X). Proof Suppose that F ∈ A(T , X) is convex-valued. By Theorem 2.3.5, one has co S(F ) = S(co F ) = S(F ). Then S(F ) is convex. If S(F ) is a convex subset of Lp (T , F, μ, X), then co S(F ) = S(F ). By Theorem 2.3.5, we have co S(F ) = S(co F ). Then S(F ) = S(co F ), which by Corollary 2.3.1, implies that F (t) = co F (t) for a.e. t ∈ T . Thus, F is convex-valued.   By the definition of p-integrable boundedness of a multifunction F : T → P(X) ¯ (t), {0}) ≤ k(t) for a.e. t ∈ T , there exists k ∈ Lp (T , R+ ) such that F (t) = : h(F where F (t) = sup{|u| : u ∈ F (t)}. If p = 1, for simplicity we shall say that F is integrably bounded, and if p = 2, we shall say that F is square integrably bounded. It is clear that F is p-integrably bounded if and only if the function T  t → F (t) ∈ R+ belongs to Lp (T , R+ ). It is clear that every measurable p-integrably bounded multifunction F : T → Cl(X) belongs to A(T , X). Remark 2.3.3 If a separable Banach space (X, | · |) has the Radon–Nikodym property, then immediately from the definition of subtrajectory integrals, it follows that for every convex-valued measurable and p-integrably bounded multifunction F : T → Cl(X) its subtrajectory integrals S(F ) is a nonempty convex weakly sequentially compact subset of Lp (T , X), which by Eberlein–Smulian theorem, implies that it is a weakly compact convex subset of this space for p > 1.    p Corollary 2.3.4 For every F ∈ A(T , X),  one hasp sup{ T |f (t)| μ(dt) : f ∈ p S(F )} = T sup{|x| : x ∈ F (t)}μ(dt) = T F (t) μ(dt). Then F is p-integrably bounded if and only if S(F ) is a bounded subset of Lp (T , X).   Theorem 2.3.6 If F, G ∈ A(T , X), then S(F + G) = S(F ) + S(G). Proof From Theorem 2.3.1 it follows that H = F + G is measurable. Indeed, by ∞ Theorem 2.3.1 there are sequences (fn )∞ n=1 ⊂ S(F ) and (gm )m=1 ⊂ S(G) such that F (t) = cl{fn (t) : n ≥ 1}, and G(t) = cl{gm (t) : n ≥ 1} for a.e. t ∈ T . Then H (t) = cl{fn (t)+gm (t) : n, m ≥ 1} for a.a. t ∈ T , which by Theorem 2.2.3 implies that H is measurable. For every f ∈ S(F ) and g ∈ S(G) one has f (t) + g(t) ∈ F (t) + G(t) for a.e. t ∈ T . But, S(H ) is closed. Therefore, S(F ) + S(G) ⊂ S(H ). ∞ On the other hand, we can select sequences (fn )∞ n=1 ⊂ S(F ) and (gm )m=1 ⊂ S(G) such that H (t) = {fn (t) + gm (t) : n, m ≥ 1}, which by Theorem 2.3.2, implies that for given h ∈ S(H ) and ε > 0 we can select a finite F-measurable partition (Ak )N k=1 of T , and positive integers n1 , . . . , nN and m1 , . . . , mN such that | h −

2.4 Notes and Remarks

N

k=1 1Ak (fnk +gmk ) |

S(F ) + S(G).

79

< ε. Hence it follows that h ∈ S(F ) + S(G). Then S(H ) ⊂  

Let (T , F, μ) be a measure space and let (X, ρ) and (Y, d) be metric spaces. If (Y, d) is separable, then a multifunction F : T × X → Cl(Y ) is said to be puniformly integrably bounded if there is k ∈ Lp (T , R+ ) such that h(F (t, x), {0}) ≤ k(t) for every x ∈ X and a.e. t ∈ T . From Theorem 2.3.4 and Remark 2.3.2 it follows that if a measurable multifunction F : T × X → Cl(Y ) is uniformly integrably bounded and F (t, ·) is h-l.s.c., then a set-valued mapping X  x → S(F (·, x)) ∈ Cl(Lp (T , Y )) is l.s.c. In the general case it does not hold true if F (t, ·) is l.s.c. But, it is also true if F (t, ·) is l.s.c. and Y is a finite-dimensional normed space. Indeed, if Y is a finite dimensional, then uniform integrable bounded multifunction F : T × X → Cl(Y ) possesses compact values. In such a case F (t, ·) is h-l.s.c. if it is l.s.c. Therefore, if F (t, ·) is l.s.c., then a set-valued mapping X  x → S(F (·, x)) ∈ Cl(Lp (T , Y )) is also l.s.c.

2.4 Notes and Remarks The definitions and results of the first two sections of this chapter are mainly based on J.P. Aubin and H. Frankowska [4], Sh. Hu and N.S. Papageorgiou [27], J.P. Aubin and A. Cellina [3], M. Kisielewicz [38], W. Hildenbrand [25], and E. Klein and A. Thomson [60]. In particular, the Michael continuous selection theorem is taken from J.P. Aubin and A. Cellina [6] and M. Kisielewicz [38]. The proof of Remark 2.2.3 can be found in Hu and Papageorgiou [27]. Proofs of the Kuratowski and RyllNardzewski measurable selection theorem and Carathèodory selection theorem are taken from Hu and Papageorgiou [27]. It was proved first by Kuratowski and RyllNardzewski in [62]. The existence of Carathèodory selectors has been considered by L. Rybi´nski in [88], A. Fryszkowski in [16], and A. Kucia and A. Nowak in [61]. In particular, A. Fryszkowski proved the existence continuous selection theorem for lower semicontinuous multifunctions with closed decomposable values instead of closed convex values. The proof of Theorem 2.1.2, dealing with the existence of Lipschitz type selectors, is taken from Sh. Hu and N.S. Papageorgiou [27]. The idea of this proof has been also presented by K. Przesławski in the paper [87]. The proofs of Lemmas 2.1.1 and 2.1.2, Remarks 2.1.1 and 2.1.2 can be found in W. Hildenbrand [25] and Hu and Papageorgiou [27], respectively. Figures 2.1, 2.2, 2.3, and 2.4 are taken from P. Aubin and A. Cellina [3] and M. Kisielewicz [38]. The proof of Remark 2.2.4 can be found in Sh. Hu and N.S. Papageorgiou [27]. The definition and properties of the Hukuhara difference of sets can be found in M. Hukuhara [28]. Section 2.3 is mainly based on F. Hiai and H. Umegaki [24] and Sh. Hu and N.S. Papageorgiou [27]. In particular, proofs of Theorems 2.3.1– 2.3.3 are taken from Sh. Hu and N.S. Papageorgiou [27], whereas Lemma 2.3.1 and Theorem 2.3.4 are taken from F. Hiai and H. Umegaki [24].

Chapter 3

Decomposable Subsets of Lp (T , F , µ, X)

In this chapter selected properties of decomposable subsets of the space of all Bochner p-integrable (equivalence classes of) functions with values in a Banach space (X, | · |) are considered. Furthermore, some properties of conditional expectations of subsets of the space Lp (T , F, μ, X) and set-valued martingales are presented.

3.1 The Space Lp (T , F , µ, X) Let (T , F, μ) be a measure space, (X, | · |) a real Banach space, and p ≥ 1. Denote by Lp (T , F, μ, X) the space of all equivalence classes [x] of Bochner p-integrable functions x : T → X under the equivalence relation: “equal μ-a.e.” The Bochner integrals of [x] will be simply denoted by T x(t)μ(dt). In particular, the above space with X = Rd and X = Rd×m will be considered. Recall that a vector function x : T → X is Bochner pintegrable, or simply p-integrable, if it is strong measurable, and |x|p : T → R+ is Lebesgue integrable, where R+ = [0, ∞). In particular, if p = 1 and p = 2 we shall write integrable and square integrable instead of 1-integrable and 2-integrable, respectively. Elements [x], [y], [z] of the space Lp (T , F, μ, X) will be simply denoted by x, y, z, respectively. It can be verified that Lp (T , F, μ, X) is a linear space with the usual operations addition “+” and multiplication “·” by scalars. It can be verified that it is a normed space with the norm | · | defined by | x | = ( T |x(t)|p μ(dt))1/p for x ∈ Lp (T , F, μ, X). If needed, the norm | · | on the space Lp (T , F, μ, X) will be denoted by | · | p to identify it with this space. The following inequalities, known as H¨older and Minkowski inequalities, respectively, can be applied to verify that the above-defined functional | · | : Lp (T , F, μ, X) → R+ possesses properties of the norm (see Section 1.2 of Chapter 1).

© Springer Nature Switzerland AG 2020 M. Kisielewicz, Set-Valued Stochastic Integrals and Applications, Springer Optimization and Its Applications 157, https://doi.org/10.1007/978-3-030-40329-4_3

81

3 Decomposable Subsets of Lp (T , F , μ, X)

82

Lemma 3.1.1 If p > 1, q = p/(p − 1), x ∈ Lp (T , F, μ, X), and y ∈ Lq (T , F, μ, X), then the following H¨older inequality 1/p 



 |x(t)||y(t)|μ(dt) ≤

1/q

|x(t)| μ(dt)

|y(t)| μ(dt)

p

T

q

T

(3.1.1)

T

is satisfied. Proof By the inequality |ξ η| ≤ (1/p)|ξ |p + (1/q)|η|q for ξ, η ∈ R it follows |ξ |1/p |η|1/q ≤ (1/p)|ξ | + (1/q)|η|. If min( T |x(t)|p μ(dt), T |y(t)|q μ(dt)) = 0, then an appropriate function is equal to zero a.e. Then (3.1.1) is satisfied. If  min( T |x(t)|p μ(dt), T |y(t)|q μ(dt)) > 0, then taking x1 (t) = |x(t)|/ | x | p , and y1 (t) = |y(t)|/ | y | q we get 

 |x(t)||y(t)|μ(dt)/(|| x | p | y | q ) = T

x1 (t)y2 (t)μ(dt) ≤ T



 |x1 (t)| μ(dt) + (1/q)

|y1 (t)|q μ(dt) = (1/p) + (1/q) = 1,

p

(1/p) T

T

 

which implies that (3.1.1) is satisfied. Lp (T , F, μ, X)

Lemma 3.1.2 If p ≥ 1, then for every x, y ∈ a function x + y belongs to Lp (T , F, μ, X), and the following Minkowski inequality



1/p |x(t) + y(t)|p μ(dt)



1/p

≤

|x(t)|p μ(dt)

T



1/p

+

T

|y(t)|p μ(dt) T

(3.1.2)

is satisfied. Proof If p = 1, then |x(t) + y(t)| ≤ |x(t)| + |y(t)| for t ∈ T . Hence, by integrating this inequality the inequality (3.1.2) follows. For p > 1 and t ∈ T one has |x(t) + y(t)|p ≤ (|x(t)| + |y(t)|)p ≤ 2p (|x(t)|p + |y(t)|p ), which implies that x + y ∈ Lp (T , F, μ, X). By H¨older inequality we get 

 |x(t) + y(t)|p μ(dt) ≤ T

(|x(t)| + |y(t)|)|x(t) + y(t)|p−1 μ(dt) = T



 |x(t)||x(t) + y(t)|p−1 μ(dt) +

T

|y(t)||x(t) + y(t)|p−1 μ(dt) ≤ T



1/p  |x(t)| μ(dt)

T

1/q |x(t) + y(t)|

p

q(p−1)

μ(dt)



1/p 

1/q

|y(t)| μ(dt)

|x(t) + y(t)|

p

T

+

T

T

q(p−1)

μ(dt)

=

3.1 The Space Lp (T , F , μ, X)

' 

1/p |x(t)| μ(dt) p

T

83

1/p ( 

 +

1/q

|y(t)| μ(dt)

|x(t) + y(t)|p μ(dt)

p

T

,

T

which implies that (3.1.2) is also satisfied for p > 1.

 

Theorem 3.1.1 (Lp (T , F, μ, X), | · |) is a Banach space for every p ≥ 1. Proof It was noted above that Lp (T , F, μ, X) is a linear space with the usual operations addition “+” and multiplication “·” by scalars. From the definition of the norm | · |, and Lemma 3.1.2 it follows that Lp (T , F, μ, X) is a normed space. We shall verify that Lp (T , F, μ, X) is a complete metric space with a metric ρ defined by the norm | · |. By Lemma 1.2.1 of Chapter 1, it is enough to verify that for every  sequence (xn ) of Lp (T , F, μ, X) such that | xn | ≤ 2−n , p it follows that a series ∞ n=1 xn converges in the space L (T , F, μ, X). Let (xn ) be a sequence of Lp (T , F, μ, X) such that | xn | ≤ 2−n . Let sn (t) = x1 (t) + . . . + xn (t), and yn (t) = |x1 (t)| + . . . + |xn (t)| for n ≥ 1 and e.e. t ∈ T . One has T |yn (t)|p μ(dt) = | |x1 | + . . . + |xn | | p and ( | x1 | + . . . + | xn | )p ≤ (1 + 1/2 + . . . + 1/2n )p ≤ 2p . Hence, it follows that limn→∞ |yn (t)|p exists for a.e. t ∈ T , because (|yn (t)|p )∞ n=1 is non-negative and non-decreasing.  Thus, limn→∞ yn (t) exists for a.e. t ∈ T , which implies that the series ∞ n=1 xn (t) converges to a finite sum x(t) for a.e. t ∈ T . By Fatou’s lemma, it follows that   p μ(dt) ≤ lim inf p μ(dt), which implies |s (t) − x(t)| |s (t) − s (t)| m→∞ T n m T n  p that a series ∞   n=1 xn converges in the space L (T , F, μ, X). Remark 3.1.1 It can be verified (see [2], 1.7 of Chap. IV) that Lp (T , F, μ, X) is separable if both a Banach space (X, | · |) and a measure space (T , F, μ) are separable. Recall (see [20], Section 40 of Chap. VIII) that the measurable space ρ) with a metric ρ defined (T , F, μ) is separable, if the Fréchet-Nikodym space (F, by ρ(A, B) = μ(A B) for A, B ∈ F is separable. Remark 3.1.2 Let (T , F, μ) be a measure space, G a sub-σ -algebra of F, and ν the restriction of μ to G. If (X, | · |) is a Banach space, then every function f ∈ Lp (T , G, ν, X) belongs to the space Lp (T , F, μ, X), and norms of the both spaces are the same. Thus, Lp (T , G, ν, X) has a natural isometric embedding in Lp (T , F, μ, X), and can be regarded as a subspace of Lp (T , F, μ, X). Remark 3.1.3 The space Lp (T , F, μ, X) can be considered with the measure μ, possessing various properties. In particular, it can be assumed that μ is finite, σ finite, non-atomic, and is a product measure μ1 × μ2 with μ1 and μ2 defined on F1 and F2 , respectively, if T = T1 × T2 and F = F1 ⊗ F2 . In the last case the space Lp (T1 × T2 , F1 ⊗ F2 , μ1 × μ2 , X) is separable if the Banach space (X, | · |), and measures spaces (T1 , F1 , μ1 ) and (T2 , F2 , μ1 2), are separable. Then for every subσ -algebra  ⊂ F1 ⊗ F2 the space Lp (T1 × T2 , , μ1 × μ2 , X) is also separable. The product measure μ1 × μ2 is non-atomic if μ1 or μ2 is non-atomic.   It is known (see [12], p. 282) that if the Banach space X∗ is separable (in particular if X is reflexive), then [L1 (T , F, μ, X)]∗ ∼ = L∞ (T , F, μ, X∗ ), i.e., the

84

3 Decomposable Subsets of Lp (T , F , μ, X)

dual Banach space of L1 (T , F, μ, X) is isomorphic and isometric by the dual pair f, f ∗ to L∞ (T , F, μ, X∗ ), where f, f ∗ = T f (t), f ∗ (t) μ(dt), for f ∈ L1 (T , F, μ, X) and f ∗ ∈ L∞ (T , F, μ, X∗ ). Much more general results (see [27], p.918) follow from the following theorems. Theorem 3.1.2 Let (T , F, μ) be a finite measure space, p > 1 and (X, | · |) be a Banach space. Then with q > 1 such that 1/p + 1/q = 1 one has [Lp (T , F, μ, X)]∗ ∼ = Lq (T , F, μ, X∗ ) if and only if X∗ has the Radon–Nikodym property.   Theorem 3.1.3 Let (T , F, μ) be a σ -finite measure space and (X, |·|) be a Banach space. Then [L1 (T , F, μ, X)]∗ ∼ = L∞ (T , F, μ, X∗ ) if and only if X∗ has the Radon–Nikodym property.   From Theorem 3.1.1, and remarks dealing with duality pairs of linear spaces presented in Chapter 1, it follows that if X is reflexive, then a weak topology of Lp (T , F, μ, X) with 1 < p < ∞ can be obtained, by taking as a basis weak topology of Lp (T , F, μ, X), weak neighborhoods of the form Vε (g) = {h ∈ Lp (T , F, μ, X) : | h − g, uk | < ε, k = 1, 2, . . . , n}, for g ∈ Lp (T , F, μ, X) with uk ∈ Lq (T , F, μ, X∗ ) for k ∈ {1, 2, . . . , n}, n ≥ 1, ε > 0 and q > 1 such that 1/p + 1/q = 1. Corollary 3.1.1 If p > 1 and (X, | · |) is a reflexive Banach space, then Lp (T , F, μ, X) is a reflexive Banach space. In particular, Lp (T , F, μ, Rd ) and Lp (T , F, μ, Rd×m ) are reflexive. Proof By Theorem 3.1.2, one has [Lp (T , F, μ, X)]∗ ∼ = Lq (T , F, μ, X∗ ). p ∗∗ q ∗ ∗ ∼ ∼ Therefore, [L (T , F, μ, X)] = [L (T , F, μ, X )] = Lp (T , F, μ, X∗∗ ) ∼ = Lp (T , F, μ, X). Then [Lp (T , F, μ, X)]∗∗ ∼ = Lp (T , F, μ, X). By the reflexivity of Rd and Rd×m , it follows that Lp (T , F, μ, Rd ) and Lp (T , F, μ, Rd×m ) are reflexive.   In what follows, we shall consider a mapping J defined on the space  Lp (T , F, μ, X) by setting J (f ) = T f (t)μ(dt) for every f ∈ Lp (T , F, μ, X). For every fixed set A ∈ F, the restriction of a mapping J to the space Lp (A, FA , μA , X) is denoted by JA , where FA = {A ∩ C : C ∈ F} and μA denote the restriction of the measure μ to FA . From properties of Bochner integrals, it follows that for every  set A ∈ F of finite measure μ(A), JA is a linear continuous mapping, because | T f (t)μ(dt)| ≤ [μ(T )]1/p | f | q with q ≥ 1 such that 1/p + 1/q = 1. Theorem 3.1.4 If p > 1 and (X, | · |) is a reflexive Banach space, then every weakly closed bounded subset of the space Lp (T , F, μ, X) is weakly compact. In particular, it holds true for a given probability space (, F, P ) and X = L2 (, F, P , Rd ).

3.1 The Space Lp (T , F , μ, X)

85

Proof The result follows immediately from Corollary 1.2.6 of Chapter 1. In particular, it is true for X = Lp (, F, P , Rd ) with p > 1, because Lp (, F, P , Rd ) is a reflexive Banach space.   Remark 3.1.4 It can be verified that every weakly closed integrably bounded subset of the space L1 (, F, P , Rd ) is weakly compact.   Theorem 3.1.5 If 1 < p < ∞, then a sequence (xn ) of the space Lp (T , F, μ, Rr ), p r with r ∈ {d, d × m} is weakly  convergent tox ∈ L (T , F, μ, R ) if and only if supn≥1 | xn | < ∞ and lim E xn (t)μ(dt) = E x(t)μ(dt), for every set E ∈ F of finite measure. Proof It is enough to observe that the above conditions are equivalent to limn→∞ x ∗ (xn ) = x ∗ (x) for every x ∗ ∈ [Lp (T , F, μ, Rr )]∗ .   Remark 3.1.5 In a similar way it can be proved (see [2], Th. 3.9 of Chap. IV) that a sequence (xn ) of the space L1 (T , F, μ, Rr ) is weakly convergent to x ∈  1 (T , F, μ, Rr ) if and only if sup | x | < ∞, and lim x (t)μ(dt) = L n n→∞ n n≥1 E  x(t)μ(dt) for every set E ∈ F of σ -finite measure.   E The following theorem is concerned with the relationship between the integration in a product measure space, and the integration in its components. Theorem (Fubini) Let (T , F, μ) and (S, , ν) be two σ -finite measure spaces, and f ∈ L1 (T × S, F ⊗ , μ × ν, X). Then for all t ∈ T , a  μ-almost  1 (S, , ν, X), and function f (t, ·) : S → X is in L { f (t, s)ν(ds)}μ(dt) = T S  f (r)(μ × ν)(dr).   T ×S Next theorem presents the relation between the theory of product measures and the theory of vector-valued integrals. Theorem 3.1.6 Let (T , F, μ) and (S, , ν) be measure spaces, which are both finite or both positive and σ -finite, and let (Z, T , λ) be their product, i.e., Z = T × S, T = F ⊗ , and λ = μ × ν. Let p ≥ 1 and F be a vector-valued μmeasurable function defined on (T , F, μ) with values in the space Lp (S, , ν, X), where X is a real Banach space. Then there is a λ-measurable function f : Z → X, which is uniquely determined except for a set of λ-measure zero, and such that f (t, ·) = F (t) for μ-almost all t ∈ T. Moreover f (t. ·) is ν-integrable on S for μ-almost all t ∈ T ,and the integral T f (t, s)μ(dt), as a function of s ∈ S, is equal to the element T F (t)μ(dt) of Lp (S, , ν, X).   Remark 3.1.6 In what follows, apart from the space Lp (T , F, μ, X) we shall consider the space Cl(Lp (T , F, μ, X)) of all nonempty closed bounded subsets of Lp (T , F, μ, X) with the Hausdorff metric H defined such as above, i.e., defined by setting H (A, B) = max{H¯ (A, B), H¯ (B, A)} for A, B ∈ Cl(Lp (T , F, μ, X)), where H¯ (A, B) = sup{dist(a, B) : a ∈ A} and H¯ (B, A) = sup{dist(b, A) : b ∈ B}.  

3 Decomposable Subsets of Lp (T , F , μ, X)

86

3.2 Decomposable Subsets of Lp (T , F , µ, X) We shall now consider properties of decomposable subsets of the space Lp (T , F, μ, X), defined in Section 2.3 of Chapter 2. Let us note that such subsets are defined in a similar way as convex subsets of linear vector spaces and therefore, decomposability seems to be as substitution of convexity in some sense. More precisely, a set K ⊂ Lp (T , F, μ, X) is defined to be F-decomposable or simply decomposable, if for every x, y ∈ K and A ∈ F one has 1A x + (1 − 1A )y ∈ K, where 1 = 1T , and 1T and 1A are characteristic functions of T and A, respectively. But 1 = 1T = 1A +1A∼ , where A∼ = T \A. Therefore, a set K ⊂ Lp (T , F, μ, X) is defined to be decomposable if for every x, y ∈ K and A ∈ F one has 1A x + 1A∼ ∈ K. It is natural to expect that properties of convex sets can be led to suitable properties of decomposable sets. In particular, it is clear that the intersection of an arbitrary family of decomposable subsets of Lp (T , F, μ, X) is decomposable. Furthermore, if (K n )n≥1 is an increasing sequence of decomposable subsets of Lp (T , F, μ, X), then ∞ n=1 Kn is decomposable. Remark 3.2.1 The simplest example of a decomposable set in Lp (T , F, μ, X) is a set S(U ) defined for a nonempty set U ⊂ X by setting S(U ) = {x ∈ Lp (T , F, μ, X) : x(t) ∈ U for a.e. t ∈ T }. An important family of decomposable sets are represented by subtrajectory integrals of measurable multifunctions F : T → Cl(X). Remark 3.2.2 The open and closed balls of the space Lp (T , F, μ, X) with the finite measure μ are examples of non-decomposable subsets of this space. Indeed, for simplicity let us consider the closed ball of the space Lp (T , F, μ, R). For the proof that it is not a decomposable subset of Lp (T , F, μ, R), it is enough to show that for every closed ball B(x0 , ρ) of Lp (T , F, μ, R), there are A ∈ F and x, y ∈ B(x0 , ρ) such that 1A x + 1A∼ y ∈ B(x0 , ρ). Select A ∈ F such that μ(A) = μ(A∼ ) = (1/2)p and let x = x0 + (3ρ/2)1A and y = x0 + (ρ/2)1A∼ . One has



| x − x0 | = A

3ρ 2

p

1/p μ(dt)

=

3ρ < ρ. 4

In a similar way, we obtain | y − x0 | < ρ. On the other hand, we have



| (1A x + 1

A∼

y) − x0 | = T

3ρ 2

p

1/p μ(dt)

=

3ρ . 2

Then there are x, y ∈ B(x0 , ρ) and A ∈ F such that 1A x + 1A∼ y ∈ B(x0 , ρ).

 

In a similar way as in the theory of convex sets, we obtain the following results dealing with properties of decomposable subsets of the space Lp (T , F, μ, X). Lemma 3.2.1 Let x1 , . . . , xn be points of a decomposable set K Lp (T , F, μ, X). ⊂ n n Then for every F-measurable partition (Ai )i=1 of T , one has i=1 1Ai xi ∈ K.

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87

Proof For n = 2 the result follows from the definition of decomposable sets. Assume that the resultis true for n = m. Then for every F-measurable partition m m+1 (Cj )m j =1 1Cj xj ∈ K. Let (Ai )i=1 be an arbitrary F-measurable j =1 of T one has m+1 partition of T and let observe that there is an F-measurable partition (Bk )k=1 of us m T such that Ai = ( k=1 Bk ) ∩ Ci for i = 1, . . . , m, and Am+1 = Bm+1 . But, for every x1 , x2 , . . . , xm+1 ∈ K, and the partition {A1 , A2 , . . . , Am+1 } of T , defined above, we have

m+1  k=1

1Ak xk =

m 

m  1(mk=1 Bk )∩Ci xi + 1Bm+1 xm+1 = 1mk=1 Bk ( 1Cj xj ) + 1Bm+1 xm+1 , j =1

i=1

and ⎛ 1mk=1 Bk ⎝

m 

⎞ 1Ci xj ⎠ + 1Bm+1 xm+1 ∈ K,

j =1

 m because xm+1 , m j =1 1Ci xj ∈ K, and { k=1 Bk , Bm+1 } is a partition of T . Therefore, by the inductive procedure, for every x1 , x 2 , . . . , xn ∈ K and every Fmeasurable partition {A1 , A2 , . . . , An } of T , we have nk=1 1Ak xk ∈ K.   Lemma 3.2.2 If K and H are decomposable subsets of Lp (T , F, μ, X), then α · K + β · H is decomposable for every α, β ∈ R. Proof If x, y ∈ α · H + β · H, then there are x, y ∈ K and x, ¯ y¯ ∈ H such that x = α · x + β · x¯ and y = α · y + β · y. ¯ By the decomposability of K and H one has 1A x + 1A∼ y ∈ K and 1A x¯ + 1A∼ y¯ ∈ H for every A ∈ F. Then for every x, y ∈ α · H + β · H and A ∈ F we get 1A x + 1A∼ y = α(1A x + 1A∼ y ) + β(1A x¯ + 1A∼ y) ¯ ∈ α · K + β · H. Thus, α · K + β · H is decomposable.   Lemma 3.2.3 The closure (weak closure) of a decomposable set K Lp (T , F, μ, X) is decomposable.

⊂

Proof Let ϕA (x, y) = 1A x + 1A∼ y for A ∈ F and x, y ∈ Lp (T , F, μ, X), and let us note that a set K ⊂ Lp (T , F, μ, X) is decomposable if and only if ϕA (K, K) ⊂ K, for every A ∈ F. But K × K = K × K and ϕA (·, ·) is continuous for every A ∈ F. Then ϕA (K × K) = ϕA (K × K) ⊂ ϕA (K × K) ⊂ K, whenever K is decomposable. Thus K is decomposable if K is decomposable. In a similar way, we can verify that the weak closure of a decomposable set K ⊂ Lp (T , F, μ, X) is decomposable, because ϕA (·, ·) is for fixed A ∈ F a linear continuous mapping and therefore, by Dunford–Schwartz theorem, it is continuous with respect to weak topology of Lp (T , F, μ, X), too. Repeating the above procedure, we see that the weak closure of the decomposable set K ⊂ Lp (T , F, μ, X) is also decomposable.  

3 Decomposable Subsets of Lp (T , F , μ, X)

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Lemma 3.2.4 Let (T , F, μ) be a σ -finite measure space and let (X, | · |) be a reflexive Banach space. If K is a decomposable subset of Lp (T , F, μ, X), then co K is decomposable. Proof By virtue of Corollary 1.2.2 of Chapter 1, one has co K = co[K]. By Theorem 2.3.3 of Chapter 2, there is F ∈ A(T , X) such that K = S(F ), because of Lemma 3.2.3, K is closed and decomposable. By Theorem 2.3.5 of Chapter 2, we have co S(F ) = S(co F ). Then co K = co[K] = S(co F ), which implies that co K is decomposable, because S(co F ) is decomposable.   Theorem 3.2.1 If (T , Fμ) is a σ -finite non-atomic measure  space, J : Lp (T , F, μ, X)  u → J (u) ∈ X is defined by J (f ) = T f (t)μ(dt) for f ∈ Lp (T , F, μ, X), and K ⊂ Lp (T , F, μ, X) is a decomposable set, then cl[J (K)] is a convex subset of X. If X is finite dimensional, then J (K) is a convex subset of the space X. Proof It suffices to show that for every u, v ∈ K, ε > 0 and α, β > 0, with α + β = 1, there is f ∈ K such that        α u(t)μ(dt) + β v(t)μ(dt) − f (t)μ(dt) < ε/2.  T

T

(3.2.1)

T

Indeed, if for every u, v ∈ K, ε > 0 and α, β ∈ [0, 1] with α + β = 1, there exists f ∈ K such that (3.2.1) is satisfied, then J (u), J (v) ∈ J (K) and |α J (u)+β J (v)− J (f )| < ε, which implies that α J (u)+β J (v) ∈ cl[J (K)]. Then for every elements a, b ∈ cl[J (K)] and α, β ∈ [0, 1], with α + β = 1, one has α a + β b ∈ cl[J (K)]. Let u, v ∈ K, ε > 0 and α, β ∈ [0, 1], with α + β = 1 be arbitrary, and define on a σ -algebra  F a vector-measure λ : F → X × X by setting λ(A) = ( A u(t)μ(dt), A v(t)μ(dt)) for every A ∈ F. By virtue of Uhl theorem the closure of the range of λ, i.e., the set cl[λ(F)],is a convex subset of X × X. But, λ(∅) = (0, 0) and λ(T ) = ( T u(t)μ(dt), T v(t)μ(dt)). Then αλ(T ) + βλ(∅) ∈ cl[λ(F)].   Therefore, for every ε >  0 there is an  A ∈ F such that |α T u(t)μ(dt)− A u(t)μ(dt)| < ε/2 and |α T v(t)μ(dt)− A v(t)μ(dt)| < ε/2. But for every A ∈ F, ε > 0 and α, β > 0, with α + β = 1, we have        α ≤ ∼ u(t)μ(dt) + β v(t)μ(dt) − (1 u + 1 v)μ(dt) A A   T

T

T

            α   u(t)μ(dt)− u(t)μ(dt) + (1−α) v(t)μ(dt)− (1−1A )v(t)μ(dt) ≤  T

A

T

T

       ε/2+  v(t)μ(dt)−α v(t)μ(dt)− v(t)μ(dt)+ 1A v(t)μ(dt) ≤ ε/2+ε/2 = ε. T

T

T

T

3.2 Decomposable Subsets of Lp (T , F , μ, X)

89

By the decomposability of the set K one has 1A u + 1A∼ v ∈ K. Taking f = 1A u + 1A∼ v, we get f ∈ K such that (3.2.1) is satisfied. In case of a finite-dimensional space X, we can repeat the above procedure with Lyapunov’s theorem.   We shall now show that every bounded decomposable weakly closed set K ⊂ Lp (T , F, μ, Rd ) with a non-atomic measure μ is convex. It will be concluded from the following result. Lemma 3.2.5 Let (T , F, μ) be a σ -finite non-atomic measure space, and let F : T → Cl(Rd ) be a measurable p-integrably bounded multifunction for p ≥ 1. Then S(co F ) = [S(F )]w , where [S(F )]w denotes the closure of the set S(F ) with respect to a weak topology of the space Lp (T , F, μ, Rd ). Proof Consider first the case p = 1. Let us note (see Lemma 1.3.3 of Chapter 1) ¯ (t), {0}). Then the multifunction that co F (t) ≤ F (t), where F (t) = h(F co F defined by (co F )(t) = co[F (t)] for t ∈ T is integrably bounded, which implies (see Theorem 1.2.3 of Chapter 1) that a set S(co F ) is a weakly compact subset of the space L1 (T , F, μ, Rd ). Then [S(F )]w ⊂ S(co F ). Let JA : L(T , F, μ, Rd ) → Rd be a linear mapping, defined for every fixed A ∈ F by setting JA (f ) = A f (t)μ(dt). By Theorem 3.2.1, JA (S(F )) is a compact convex subset of Rd for every A ∈ F. Then JA (S(F )) = co[JA (S(F ))] = cl[JA (S(co F ))] = JA (S(co F )). Therefore, for every u ∈ S(co F ) and A ∈ F Nm there exists vA ∈ S(F ) such that JA (u) = JA (vA ). Let (Am j )j =0 be for every m ≥ 1  an F-measurable partition of T such that Am F (t)μ(dt) ≤ 2−j /(4 m) for every j

m ≥ 1 and 0 ≤ j ≤ Nm . Select now for every u ∈ S(co F ), every m ≥ 1, and 0 ≤ j ≤ Nm , an element vjm ∈ S(F ) such that JAmj (u) = JAmj (vjm ), and let  m m m vm = N j =0 1Aj vj . By the decomposability of S(F ), for every m ≥ 1 one has vm ∈ S(F ). Then vm (t) ∈ F (t) for a.e. t ∈ T . Therefore, for m ≥ 1 and a.e. t ∈ T we have |vm (t)| ≤ F (t), which implies that (vm )∞ m=1 is the relatively weakly compact sequence of the set S(F ). Then there exists v ∈ [S(F )]w , and a subsequence if needed, such that vm  v as m → ∞, where vm  v denotes the weak convergence of a sequence (vm )∞ m=1 to v. On the other hand, for every m ≥ 1, and A ∈ F one has    Nm       =  vm (t)μ(dt) − u(t)μ(dt) 1Amj (t)vjm (t)μ(dt)    A

A

−

A j =0

  Nm A j =0

Nm       j =0

A∩Am j

 

 

vjm (t)μ(dt) +  

  1Amj (t)u(t)μ(dt) ≤

A∩Am j

 Nm     u(t)μ(dt) ≤ 2 

j =0

Am j

F (t)μ(dt) ≤

1 . m

Therefore, limm→∞ A vm (t)μ(dt) = A u(t)μ(dt) for every A ∈ F of the finite measure μ(A), which by Remark 3.1.5 implies that vm  u as m → ∞. Then

90

3 Decomposable Subsets of Lp (T , F , μ, X)

u − v and therefore, for every u ∈ S(co F ) one has u ∈ [S(F )]w . Thus, S(co F ) ⊂ [S(F )]w . In the similar way, for p > 1 we also get S(co F ) = [S(F )]w .   Theorem 3.2.2 Let (T , Fμ) be a σ -finite non-atomic measure space, and let K ⊂ Lp (T , F, μ, Rd ) be a bounded weakly closed decomposable set. Then K is a convex subset of the space Lp (T , F, μ, Rd ). Proof Let us note that K is a closed and decomposable subset of the space Lp (T , F, μ, Rd ). Then by Theorem 2.3.3 of Chapter 2, there is a measurable multifunction F : T → Cl(Rd ) such that K = S(F ). By boundedness of K a multifunction F (see Corollary 2.3.4 of Chapter 2) is p-integrably bounded. By virtue of Lemma 3.2.5, one has S(co F ) = [S(F )]w = [K]w = K. Thus, K is convex.   Remark 3.2.3 The above result can be extended (see [27], Th. 3.17 of Chap. 2) for bounded weakly closed decomposable subsets of the space Lp (T , F, μ, X), where X is a Banach space.   We shall now show that a bounded decomposable subset of the space Lp (T , F, μ, X) with a finite measure μ is uniformly integrable. Let us p recall that a set K  ⊂ Lp (T , F, μ, X) is said to be uniformly integrable if limμ(E)→0 supg∈K E |g(t)| μ(dt) = 0. It is clear that every subset of the space Lp (T , A, μ, X) is uniformly integrable if it is integrably bounded. Recall, that a set K ⊂ Lp (T , A, μ, X) is integrably bounded, if there exists a function m ∈ Lp (T , F, μ, R) such that |f (t)| ≤ m(t) for every f ∈ K and a.e. t ∈ T . In such case a set K is a bounded subset of the space Lp (T , F, μ, X). Lemma 3.2.6 If (T , F, μ) is a finite measure space, and R is a family of R+ -valued measurable functions, then there exists a (μ-a.e. unique) measurable function h : T → R+ such that: (1) f (t) ≤ h(t) a.e. for every f ∈ R and (2) if g : T → R+ is measurable function such that f (t) ≤ g(t) a.e. for every f ∈ R, then h(t) ≤ g(t) a.e. Moreover, there exists a sequence (fn )∞ n=1 ⊂ R such that h(t) = supn≥1 fn (t) a.e. Finally, if R is directed upwards, i.e., if R is such that for every f1 , f2 ∈ R, there exists f3 ∈ R such that max{f1 (t), f2 (t)} ≤ f3 (t) a.e., then the sequence (fn )∞ n=1 can be chosen to be increasing. Proof Without any loss of generality, by considering the increasing homeomorphism ϕ(z) = z/(1 + z) mapping R+ onto [0, 1], we may assume that the functions in R take their values in [0, 1]. Let C be the class of all countable subfamilies of R. For every set G ∈ C, let us define fG = sup{f : f ∈ G}. It is clear that  fG is measurable. Let m = sup{ T fG (t)μ(dt) : G ∈ C}. This supremum is attained, because if (Gn )∞ of C such that T fGn (t)μ(dt) → m. n=1 is a sequence   Then T fG (t)μ(dt) = m with G = = h n≥1 Gn ∈ C. We shall show that fG ∪ {f } ∈ C. Then is the desired maximum function. Let f ∈ R, and G = G   fG = max{h, f } and so m = T h(t)μ(dt) ≤ T max{h(t), f (t)}μ(dt) ≤ m < ∞. Thus, h(t) = max{h(t), f (t)} a.e. and so f (t) ≤ h(t) a.e. Also, from the definition ∞ of h = fG , it is clear that h = supn≥1 fn for some sequence (fn )n=1 ⊂ R, and

3.3 Decomposable Hulls of Subsets of Lp (T , F , μ, X)

91

this also proves the uniqueness μ-a.e. of h. Finally, if R is directed upwards, we  can get h = supn≥1 fn with (fn )∞ n=1 increasing a.e., by taking f1 = f1 , and  ∞   fn ≥ max{fn−1 , fn } a.e. Of course we have (fn )n=1 ⊂ R.   We prove now the desired result. Theorem 3.2.3 Let (T , F, μ) be a finite measure space, and let (X, | · |) be a Banach space. If K is a bounded decomposable subset of the space Lp (T , F, μ, X), then it is uniformly integrable. Proof Let |K| = {|f | : f ∈ K}, where |f | is a function defined by |f |(t) = |f (t)| for t ∈ T . We have |K| ⊂ Lp (T , F, μ, R+ ). Let h = ess sup|K|, where ess sup|K| denotes the least upper bound of the family |K| in the sense of in equality a.e. By Lemma 3.2.6, we infer that there exists a sequence (fn )∞ n=1 ⊂ K such that h(t) = supn≥1 |fn (t)| a.e. Moreover, the decomposability of K implies that |K| is directed upwards. Thus we can choose the increasing sequence (fn )∞ n=1 such that |fn | → h a.e. Since K is bounded, an application of the monotone convergence theorem implies that h ∈ Lp (T , F, μ, R). But for all f ∈ K one has |f (t)| ≤ h(t) a.e. Then K is uniformly integrable.   Remark 3.2.4 It can be verified (see [13], Theorem 2.1 of Chap. IV) that if X is reflexive, then every bounded decomposable set in Lp (T , F, μ, X) is relatively weakly compact.  

3.3 Decomposable Hulls of Subsets of Lp (T , F , µ, X) We shall now consider some properties of decomposable and closed decomposable hulls of subsets of the space Lp (T , F, μ, X). Recall (see Section 2.3 of Chapter 2) that for a given set K ⊂ Lp (T , F, μ, X) the decomposable hull dec K and the closed decomposable hull dec(K) of K are defined to be the smallest decomposable and closed decomposable sets, respectively, containing a set K. We begin with some representation theorem for decomposable subsets of the space Lp (T , F, μ, X). From Theorem 2.3.3 of Chapter 2 and Corollary 2.3.2 of Chapter 2, the following results follow. Theorem 3.3.1 For every nonempty decomposable set K ⊂ Lp (T , F, μ, X) there exists a sequence (gn )∞ n=1 ⊂ K such that K = dec{gn : n ≥ 1}, where the closures are taken with respect to the norm topology of Lp (T , F, μ, X). Proof Let K be a decomposable subset of Lp (T , F, μ, X). By Lemma 3.2.3, the closure K of the set K is decomposable. Then by virtue of Theorem 2.3.3 of Chapter 2, there exists a multifunction G ∈ A(T , X) such that K = S(G). By Corollary 2.3.2 of Chapter 2, there exists a sequence (gn )∞ n=1 ⊂ S(G) such that G(t) = cl{gn (t) : n ≥ 1} for t ∈ T and such that S(G) = dec{gn : n ≥ 1}. Then there is a sequence (gn )∞   n=1 ⊂ K such that K = dec{gn : n ≥ 1}.

3 Decomposable Subsets of Lp (T , F , μ, X)

92

Given a nonempty decomposable set K ⊂ Lp (T , F, μ, X) a sequence (gn )∞ n=1 ⊂ K such that K = dec{gn : n ≥ 1} is called representation Castaing of a decomposable set K. Theorem 3.3.2 Let K be a nonempty decomposable subset of the space Lp (T , F, μ, X). For every representation Castaing (gn )∞ n=1 of K, and every F ⊗ β(X)-measurable function φ : T × X → [−∞, +∞] such that φ(t, ·) is upper semicontinuous and such that T (φ ◦ g0 )(t)μ(dt) < ∞ for some g0 ∈ K,  one has infg∈K J (φ ◦ g) = T infn≥1 (φ ◦ gn )(t)μ(dt), where J (f ) = T f (t)μ(dt) and (φ ◦ f )(t) = φ(t, f (t)) for every f ∈ Lp (T , F, μ, X) and t ∈ T . Proof By Theorem 2.3.3 of Chapter 2, there exists a multifunction F ∈ A(T , X) such that K = S(F ). By Theorem 2.3.4 of Chapter 2, one has inf{J (φ ◦ f ) : f ∈ S(F )} = T inf{φ(t, x) : x ∈ F (t)}μ(dt). Then  infg∈K J (φ ◦ g) = inf{J (φ ◦ f ) : f ∈ K)} = inf{J (φ ◦ f ) : f ∈ S(F )} = T inf{φ(t, x) : x ∈ F (t)}μ(dt). Let (gn )∞ n=1 ⊂ S(F ) be such that F (t) = cl{gn (t) : n ≥ 1} for t ∈ T and S(F ) = dec{gn : n ≥ 1}. We have  inf J (φ ◦ g) =

g∈K

inf{φ(t, x) : x ∈ F (t)}μ(dt) = T



 inf{φ(t, x) : x ∈ {gn (t) : n ≥ 1}}μ(dt) = T

Then infg∈K J (φ ◦ g) =

 T

infn≥1 (φ ◦ gn )(t)μ(dt).

inf (φ ◦ gn )(t)μ(dt).

T n≥1

 

Remark 3.3.1 Similarly it can be verified that for a given nonempty decomposable set K ⊂ Lp (T , F, μ, X), every representation Castaing (gn )∞ n=1 of K and every F ⊗ β(X)-measurable function φ : T × X → [−∞, +∞] such that φ(t, ·) is upper  semicontinuous and such that T (φ ◦ g0 )(t)μ(dt) > −∞ for some g0 ∈ K, one has supg∈K J (φ ◦ g) = T supn≥1 (φ ◦ gn )(t)μ(dt).   Remark 3.3.2 The above results are also satisfied if (T , F, μ) is complete and φ : T × X → [−∞, +∞] is lower semicontinuous with respect x ∈ X for fixed t ∈ T  instead upper semicontinuous, and T (φ ◦ g0 )(t)μ(dt) < ∞ for some g0 ∈ K or    T (φ ◦ g0 )(t)μ(dt) > −∞ for some g0 ∈ K. Lemma 3.3.1 For a given set ⊂ Lp (T , F, μ, X), a set dec(K) consists of all K n decomposable combinations i=1 1Ai xi with x1 , . . . , xn ∈ K, and every finite Fmeasurable partitions (Ai )ni=1 of T .  Proof Let D be the family of all decomposable combinations ni=1 1Ai xi with x1 , . . . , xn ∈ K and every finite F-measurable partitions (Ai )ni=1 of T . Let us note that D is a decomposable set containing K. Indeed, for every x, y ∈ D there are x1 , . . . , xn ∈ K, y1 , . . . , ym ∈ K, finite F-measurable partitions (Ai )ni=1 and m n m (Bj )j =1 of T such that x = i=1 1Ai xi and y = j =1 1Bj yj . By Lemma 3.2.1, it follows that x, y ∈ dec(K). Taking intersections Ai ∩ Bj of elements of the

3.3 Decomposable Hulls of Subsets of Lp (T , F , μ, X)

93

above partitions (Ai )ni=1 , and (Bj )m , we obtain the partition (Ck )rk=1 of T , such j =1 r r xk and y = yk , where xk ∈ {x1 , . . . , xn } and that x = k=1 1Ck k=1 1Ck yk ∈ {y1 , . . . , ym } for k = 1, . . .  , r. Therefore, 1A x + 1A∼ y ∈ D for every 2r k : k = 1, 2, . . . 2r} = A ∈ F, because 1A x + 1A∼ y = k zk with {C k=1 1C ∼ {Ck ∩ A, Ck ∩ A : k = 1, 2, . . . , r} and zk ∈ { x1 , . . . , xr } ∪ { y1 , . . . , yr } ⊂ K. For every x ∈ K and z ∈ K one has x = 1T x + 1∅ z ∈ D. Then K ⊂ D, which by the decomposability of D implies that dec(K) ⊂ D. On the other hand, by the definition of a family D and Lemma 3.2.1, for every x ∈ D one has x ∈ dec(K). Then D ⊂ dec(K). Therefore, D = dec(K).   Lemma 3.3.2 For a given set K ⊂ Lp (T , F, μ, X), one has dec(K) = cl[dec(K)], and dec(K) = dec(K). Proof It was noted above that dec(K) is a closed decomposable subset of the space Lp (T , F, μ, X), because dec(K) is the intersection of closed decomposable subsets of the space Lp (T , F, μ, X) containing K. Hence it follows that cl[dec(K)] ⊂ dec(K). On the other hand, by Lemma 3.2.3, cl[dec(K)] is a closed decomposable set containing K. Therefore, dec(K) ⊂ cl[dec(K)]. Then dec(K) = cl[dec(K)]. The equality dec(K) = dec(K) can be verified similarly to Corollary 1.2.2 of Chapter 1. Indeed, it is clear that dec(K) ⊂ dec(K). On the other hand, for every ε > 0 and x ∈ dec(K) = cl[dec(K)] there exists zε ∈ dec(K), such that | x − zε | < ε/2. By Lemma 3.3.1, there exist x1 , . . . , xn ∈ K and an F-measurable partition (Ai )ni=1 of T such that zε = ni=1 1Ai xi . For every i = 1, 2, . . . , n there  exists xi ∈ K such that | xi − xi | 2 < [ε/(2n)]2 . Therefore, | zε − ni=1 1Ai xi | 2 =  n n n 2 2 | i=1 1Ai (xi − xi ) | = T | i=1 1Ai (xi − xi )| μ(dt) = i=1 Ai |(xi −   xi |2 μ(dt) ≤ ni=1 | xi − xi | 2 < ε2 /4. But, | x − zε | 0 there exists a finite F-measurable partition (Ai )ni=1 of n T , and a family {x1 , . . . , xn } ⊂ K such that | x − i=1 1Ai xi | ≤ ε.   Lemma 3.3.3 If K is a convex subset of the space Lp (T , F, μ, X), then dec(K) and dec(K) are convex decomposable subsets of this space. Proof The decomposability of dec(K) and dec(K) follows immediately from the definitions of decomposable and closed decomposable hulls. Let K be a convex subset of Lp (T , F, μ, X) and u, v ∈ dec(K). By Lemma 3.3.1, there are finite FM M measurable partition (An )N of T , and families (un )N n=1 and (Bm )m=1 n=1 , (vm )m=1 ⊂ M N K, such that u = n=1 1An un and v = m=1 1Bm vm . Let (Dk )R k=1 be a finite F R uk and v = R vk with measurable partition of T such that u = k=1 1Dk k=1 1Dk uk ∈ {u1 , . . . , uN } and vk ∈ {v1 , . . . , vM } for k ∈ {1, . . . , R}. For every λ ∈ [0, 1] and k ∈ {1, . . . , R} one has λu¯ k + (1 − λ)v¯k ∈ K. Therefore, λu + (1 − λ)v =  R ¯ k +(1−λ)v¯k ] ∈ dec{K}. Thus, dec(K) is a convex subset of the space k=1 1Dk [λu

94

3 Decomposable Subsets of Lp (T , F , μ, X)

Lp (T , F, μ, X). Hence, by Lemmas 3.3.2 and 1.2.6 of Chapter 1, the convexity of dec(K) also follows.   Theorem 3.3.3 Let (X, | · |) be a Banach space with X∗ having Radon–Nikodym property. For every p ≥ 1 and every set K ⊂ Lp (T , F, μ, X), one has dec[co(K)] = co[dec(K)]. Proof It is clear that K ⊂ dec[co(K)]. Then by virtue of Lemma 3.3.3, dec[co(K)] is a closed convex decomposable set containing K. Therefore, co[dec(K)] ⊂ dec[co(K)]. To prove that dec[co(K)] ⊂ co[dec(K)] let us note that by Corollary 1.2.2 of Chapter 1, we have co[dec(K)] = co[dec(K)]. By Theorem 2.3.3 of Chapter 2, there exists F ∈ A(T , X) such that S(F ) = [dec(K)], which by Theorem 2.3.5 of Chapter 2, and Corollary 1.2.2 of Chapter 1 implies that S(co F ) = co S(F ) = co[dec(K)] = co[dec(K)]. Thus, co[dec(K)] is decomposable, because S(co F ) is decomposable. Then co(K) ⊂ co[dec(K)], which implies that dec[co(K)] ⊂ co[dec(K)].   Theorem 3.3.4 Let K and H be subsets of Lp (T , F, μ, X) and let α ∈ R. Then (i) dec(α · K) = α · dec(K) and dec(K + H) = dec(K) + dec(H), (ii) dec(α · K) = α · dec(K), (iii) if dec(K) is weakly compact (compact), and dec(H) is weakly closed (closed), then dec(K + H) = dec(K) + dec(H). Proof (i) By Lemma 3.2.2, a set α · dec(K) is decomposable, and α · K ⊂ α · dec(K). Therefore, dec(α ·K) ⊂ α ·dec(K). Let u ∈ α ·dec(K) and x ∈ dec(K) be such n that u = α·x. By Lemma 3.3.1, there is  a finite F-measurable partition n (Ai )i=1 n of T and x1 , . . . , xn ∈ K such that x = i=1 1Ai xi . Then α·x = i=1 1Ai (α· xi ), which implies that α · x ∈ dec(α · K). Thus, α · dec(K) ⊂ dec(α · K). We get dec(K + H) ⊂ dec(K) + dec(H), because K + H ⊂ dec(K) + dec(H) and by Lemma 3.2.2, the set dec(K) + dec(H) is decomposable. In a similar way as above, we obtain dec(K) + dec(H) ⊂ dec(K + H). Indeed, for every x ∈ dec(K) and y ∈ dec(H) there are partitions (Ci )ni=1 and (Di )m i=1 nof T and families {x , . . . , x } ⊂ K and {y , . . . , y } ⊂ H such that x = 1 n 1 m i=1 1Ci xi  N be a partition of T , and ( N ⊂ K and and y = m 1 y . Let (E ) x ) D i i i i i=1 i=1 i=1  ⊂ H be such that x = N 1Ei xi and y = N 1Ei yi . Then x +y = ( yi )N i=1 i=1 i=1 N xi + yi ) ∈ dec(K + H), which implies that dec(K) + dec(H) ⊂ i=1 1Ei ( dec(K + H). (ii) By (i), one has dec(α · K) = α · dec(K). Therefore, cl[dec(α · K)] = cl[α · dec(K)] = α · cl[dec(K)]. By virtue of Lemma 3.3.2, it follows that dec(α · K) = α · dec(K), because cl[dec(K)] = cl[dec(K)] = dec(K). (iii) By virtue of Corollary 1.2.4 of Chapter 1, a set dec(K) + dec(H) is weakly closed and therefore, it is closed. By Lemma 3.2.2, it is decomposable. Then dec(K) + dec(H) is a closed decomposable set containing K + H. Therefore, dec(K+H) ⊂ dec(K)+dec(H). To verify that the converse is also satisfied, let

3.3 Decomposable Hulls of Subsets of Lp (T , F , μ, X)

95

us note that by Corollary 3.3.1, for every x ∈ dec(K), y ∈ dec(H) and ε > 0 there are partitions (Ci )ni=1 and (Di )m T , and families {x1 , . . . , xn } ⊂ i=1 of  n K and {y , . . . , y } ⊂ H such that | x − 1 m i=1 1Ci xi | ≤ ε/2 and | y − m 1 y | ≤ ε/2. Taking, similarly as above, a partition (Ei )N of T , i=1 Di i  i=1 N N N N yi )i=1 ⊂ H such that x = i=1 1Ei xi and y = i=1 1Ei yi ( xi )i=1 ⊂ K and ( we get | (x + y) −

N 

1Ei ( xi + yi ) | ≤ | x −

i=1

n 

1Ci xi | + | y −

m 

i=1

1Di yi | ≤ ε,

i=1

which by Corollary 3.3.1 (Lemma 3.3.4) implies that x + y ∈ dec(K + H). Thus, dec(K) + dec(H) ⊂ dec(K + H).   Corollary 3.3.2 If (X, |·|) has the Radon–Nikodym property, then for every set K ⊂ Lp (T , F, μ, X), one has co[dec(K)] = co[dec(K)] = dec[co(K) = dec[co(K). Proof By Corollary 1.2.2 of Chapter 1, Lemma 3.3.2, and Theorem 3.3.3, we get co[dec(K) = co[dec(K)], dec[co(K) = dec[co(K)], and dec[co(K)] = co[dec(K)]. Therefore, co[dec(K)] = dec[co(K)] = dec[co(K)].   We have the following natural and intrinsic characterization of integrably bounded subsets of the space Lp (T , F, μ, X). Theorem 3.3.5 Let  be a nonempty subset of the space Lp (T , F, μ, X) and let M() = {max[|u1 |, . . . , |un |] : u1 , . . . , un ∈ , n ≥ 1}. The following statements are equivalent: (i) (ii) (iii) (iv) (v)

 is p-integrably bounded, dec  is p-integrably bounded, dec  is a bounded subset of the space Lp (T , F, μ, X), M() is a bounded subset of the space Lp (T , F, μ, R), M() is p-integrably bounded.

Proof We shall prove first that (ii) ⇒ (iii) ⇒ (iv) ⇒ (ii), next (iv) ⇔ (v) and (i) ⇔ (iii). The implication (ii) ⇒ (iii) is obvious. To prove (iii) ⇒ (iv), let us note first that     sup |ut |p μ(dt) : u ∈ dec  ≥ sup |ut |p μ(dt) : u ∈ dec{u1 , . . . , un } T

T

for every u1 , . . . , un ∈ , and n ≥ 1. But, for every n ≥ 1 one has 



|u(t)| μ(dt) : u ∈ dec{u1 , . . . , un } = p

sup T

3 Decomposable Subsets of Lp (T , F , μ, X)

96

sup

  n T



p 1Ak (t)|uk (t)|

 μ(dt) : (Ak )nk=1 ∈ (T , F) =

k=1

  v p (t)μ(dt) : v ∈ dec{|u1 |, . . . , |un |} ≥ (max{|u1 (t)|, . . . , |un (t)|})p μ(dt),

sup T

T

where (T , F) denotes the family of all finite F-measurable partitions of T . Then    |ut |p μ(dt) : u ∈ dec  ≥ sup v p (t)μ(dt) : v ∈ M() .

 sup T

(3.3.1)

T

Thus (iii)⇒(iv) is satisfied. To prove the implication (iv)⇒(v), let us note that 





|ut | μ(dt) : u ∈ dec  = sup

sup

|ut | μ(dt) : u ∈ dec  = p

T

sup



p

  n T

T

p 1Ak (t)|uk (t)|

 μ(dt) :

(Ak )nk=1

∈ (T , F ), u1 , . . . , un ∈ , n ≥ 1 ≤

k=1



 v (t)μ(dt) : v ∈ M() . p

sup T

By Theorem 2.3.3 of Chapter 2, there exists an F-measurable multifunction F : T → Cl(X) such that S(F ) = dec . Therefore, by Remark 2.3.2 of Chapter 2, we get 

 F (t) μ(dt) = sup T

 |u(t)| μ(dt) : u ∈ dec  ≤

p

p

(3.3.2)

T



 v (t)μ(dt) : v ∈ M() < ∞. p

sup T

Therefore, F is p-integrably bounded, which implies that also dec  is p-integrably bounded, because for every u ∈ dec  one has |u(t)| ≤ F (t) for e.e. t ∈ T . Then the assertion by inequalities (3.3.1) and (3.3.2)  (ii) holds true a.e. Moreover,  it follows that T F (t)p μ(dt) = sup{ T v p (t)μ(dt) : v ∈ M()}. Indeed, by Remark 2.3.2 of Chapter 2, one has    |u(t)|2 μ(dt) : u ∈ dec  = sup |u(t)|2 μ(dt) : u ∈ S(F ) =

 sup T

T



 sup{|x|2 : x ∈ F (t)}μ(dt) =

T

F (t)2 μ(dt). T

3.3 Decomposable Hulls of Subsets of Lp (T , F , μ, X)

97

 Hence, by (3.3.1) and (3.3.2) the equality T F (t)2 μ(dt) = sup{v 2 (t)μ(dt) : v ∈ M()} follows. But, |v(t)| ≤ F (t) for every v ∈ M(). Then the implication (iv) ⇒ (v) is satisfied. It is clear that we also have (v) ⇒ (iv). Finally, in order to prove that (i) ⇒ (iii) let us note first that by (ii) ⇒ (iii) and (ii) ⇒ (i) it follows that (iii) ⇒  (i). Conversely, for every u ∈ dec  and ε > 0 there is bε ∈ dec  such that T |u(t) − bε (t)|p μ(dt) ≤ ε. But  is p-integrably bounded, then there exists h ∈ Lp (T , F, μ, R+ ) such that  |γ (t)| ≤ h(t) for a.e.  t ∈ T and every γ ∈ . Then for every u ∈ dec  one has T |u(t)|p μ(dt) ≤ ε + T [h(t)]p μ(dt) < ∞.   It was verified above (see Remark 3.2.2) that the closed unit ball in the space Lp (T , F, μ, X) is non-decomposable subset of this space. It can be verified that the decomposable hull of the unit ball of the space Lp (T , F, μ, X) is equal to the whole space Lp (T , F, μ, X). Theorem 3.3.6 Let p ≥ 1, and let B be the unit ball in Lp (T , F, μ, X). Then dec(B) = Lp (T , F, μ, X). Proof Take an arbitrary u ∈ Lp (T , F, μ, X). We shall show that u ∈ dec(B). Indeed, fix an integer n ≥ 1 such that | u | p ≤ n. Let (Aα )α∈[0,1] be a segment for a real measure m(A) = A |u(t)|p μ(dt), i.e., an increasing family (Aα )α∈[0,1] of subsets of T such that A0 = ∅, A1 = T , and m(α) = α · m(T ). Let Bk = Ak/n \ A(k−1)/n for k= 1, . . . , n. The family {B1 , . . . , Bn } is a partition of T such that m(Bk )  = (1/k) T |u(t)|p μ(dt) ≤ 1. But uk = 1Bk u∈n B for k = 1, . . . , n. Therefore, nk=1 1Bk uk ∈ dec(B) and nk=1 1Bk uk = k=1 1Bk u = u. Thus Lp (T , F, μ, X) ⊂ dec(B) ⊂ Lp (T , F, μ, X), because B ⊂ Lp (T , F, μ, X) and Lp (T , F, μ, X) is decomposable. Thus, dec(B) = Lp (T , F, μ, X).   Given a probability space PF = (, F, F, P ) with a filtration F = (Ft )0≤t≤T , we shall present some properties of subsets of the space L2 ([0, T ] × , F , Rr ) defined by decomposable hulls, where F is a σ -algebra of all F-non-anticipative subsets of [0, T ] × . These sets, denoted by decπ (H), are defined for given a partition π = {0 = τ0 < τ1 < . . . < τK = T } of the interval [0, T ], and a nonempty set H ⊂ L2 ([0, T ] × , F , Rr ) by setting decπ (H) = 1[0,τ1 ] decF0 (H) + 1(τ1 ,τ2 ] decFτ1 (H) + . . . + 1(τK−1 ,T ] decFK−1 (H). We begin with the following lemma. Lemma 3.3.4 For every set K ⊂ L2 ([0, T ] × , F , Rr ) with r ∈ {d, d × m}, and every partition π = {0 = τ0 < τ1 < . . . < τp = T } of the interval [0, T ] one has K ⊂ 1[0,τ1 ] K + 1(τ1 ,τ2 ] K + . . . + 1(τp−1 ,T ] K. If K is decomposable, then K = 1[0,τ1 ] K + 1(τ1 ,τ2 ] K + . . . + 1(τp−1 ,T ] K. Proof For a given above partition π one has K = 1[0,T ] {a : a ∈ K} = {1[0,τ1 ] a + 1(τ1 ,τ2 ] a + . . . + 1(τn−1 ,T ] a : a ∈ K} ⊂ {1[0,τ1 ] a : a ∈ K} + {1(τ1 ,τ2 ] a : a ∈ K} + . . . + {1(τn−1 ,T ] a : a ∈ K} = 1[0,τ1 ] K + 1(τ1 ,τ2 ] K + . . . + 1(τp−1 ,T ] K. If K is decomposable, then for every a, b ∈ K one has 1[0,τ1 ] a + 1(τ1 ,T ] b ∈ K. Therefore, K ⊂ {1[0,τ1 ] a : a ∈ K} + {1(τ1 ,T ] a : a ∈ K} ⊂ K. Hence, by the induction procedure, it follows that K ⊂ {1[0,τ1 ] a : a ∈ K} + {1(τ1 ,τ2 ] a : a ∈ K} + . . . +

3 Decomposable Subsets of Lp (T , F , μ, X)

98

{1(τn−1 ,T ] a : a ∈ K} ⊂ K for every partition π = {0 = τ0 < τ1 < . . . < τp = T } of the interval [0, T ].   We shall now prove the following results. Lemma 3.3.5 For every partition π = {0 = τ0 < τ1 < . . . < τp = T } of the interval [0, T ], and a nonempty convex integrably bounded set H ⊂ L2 ([0, T ] × , F , Rr ) with r ∈ {d, d × m}, a set decπ (H) is a convex integrably bounded subset of the space L2 ([0, T ] × , F , Rr ). Proof From the definition of the set decπ (H), for every u ∈ decπ (H) there i i exist partitions (Aik )N ), and finite sequences (gki )N ⊂ H with k=1 ∈ (, Fτi−1 k=1 N1  i p i 1 i = 1, . . . , p such that u = 1[0,τ1 ] k=1 1A1 gk + i=2 1(τi−1 ,τi ] N k=1 1Aik gk , k where (, Fτi−1 ) denotes the set of all finite Fτi−1 -measurable partitions of . i Ni i By properties of partitions (Aik )N k=1∈ (, Fτi−1 ), and sequences (gk )k=1 ⊂ i i H, for i = 1, 2, . . . , p, a process N k=1 1(τi−1 ,τi ]×Aik gk is measurable for every i = 1, . . . , p. Furthermore, for every fixed τ ∈ (τi−1 , τi ], a random variable N i i k=1 1(τi−1 ,τi ]×Aik (τ, ·) gk (τ, ·) is Fτ -measurable for every i = 1, . . . , p. Then N1 Ni  i a process 1[0,τ1 ] k=1 1A1 gk1 + K i=2 1(τi−1 ,τi ] k=1 1Ai gk is F-non-anticipative k

k

i Ni i for every partition (Aik )N k=1 ∈ (, Fτi−1 ), and every family (gk )k=1 ⊂ H with N i i = 1, . . . , p. On the other hand, for every partition (Aik )k=1 ∈ (, Fτi−1 ), and Ni i every family (gk )k=1 ⊂ H one has

 2 p Ni N1      1 i 1[0,τ ] (t)  = 1 (ω)g (t, ω)+ 1 (t) 1 (ω)g (t, ω) i 1 (τi−1 ,τi ] 1 k k A A   k

k

k=1

1[s,τ1 ] (t)

N1 

i=2

1A1 (ω)|gk1 (t, ω)|2 k

k=1

+

p 

(3.3.3)

k=1

1(τi−1 ,τi ] (t)

i=2

Ni 

1Ai (ω)|gki (t, ω)|2 ≤ λt (ω), k

k=1

for a.e. (t, ω) ∈ R+ × , where λ : [0, T ] ×  → R+ is a square integrable process such that |g(t, ω)|2 ≤ λt (ω) for every g ∈ H and a.e. (t, ω) ∈ R+ × . Thus, decπ (H) is a square integrably bounded subset of the space L2 ([0, T ]×, F , Rr ). i i N i Finally, for every u, u ∈ decπ (H) there are partitions (Aik )N k=1 , (Ak )k=1 ∈

Ni i gki )k=1 ⊂ H such that (, Fτi−1 ), and families (gki )N k=1 , (

u = 1[0,τ1 ]

N1 

p 

k

k=1

and

1A1 gk1 +

i=2

1(τi−1 ,τi ]

Ni 

1Ai gki k

k=1

3.3 Decomposable Hulls of Subsets of Lp (T , F , μ, X)

u = 1[0,τ1 ]

N1 

1A 1 gk1 +

p 

k

k=1

99

1(τi−1 ,τi ]

Ni 

i=2

1A i gki . k

k=1

i Mi i i Mi Let a partition (Bki )M k=1 ∈ (, Fτi−1 ) and families (fk )k=1 , (fk )k=1 ⊂ H be given for i = 1, . . . , p, such that

u = 1[0,τ1 ]

M1 

p 

1B 1 fk1 + k

k=1

1(τi−1 ,τi ]

i=2

Mi 

1B i fki k

k=1

and u = 1[0,τ1 ]

M1 

1B 1 f k1 +

p 

k

k=1

1(τi−1 ,τi ]

i=2

Mi 

1B i f ki . k

k=1

Now for every α ∈ [0, 1] it follows that αu + (1 − α) u= 1[0,τ1 ]

M1 

p Mi   1B 1 [αfk1 +(1−α)f k1 ]+ 1(τi−1 ,τi ] 1B i [αfki +(1−α)f ki ] ∈ decπ (H), k

k

k=1

i=2

k=1

because αfki + (1 − α)f ki ∈ H for every i = 1, . . . , p.

 

Lemma 3.3.6 For every nonempty set H ⊂ L2 ([0, T ] × , F , Rr ), and a n sequence (πn )∞ n=1 of partitions of the interval [0, T ] of the form πn = {0 < τ1 < n n n n n τ2 < . . . < τpn −1 < τpn } with τi = iδn for i = 1, . . . , pn , where δn = T /2 and pn = 2n , one has decπn (H) ⊂ decπn+1 (H) for every n ≥ 1. Proof For every n ≥ 1, one has 1[0,τ1n ] decF0 (H) = 1[0,τ n+1 ] decF0 (H) + 1 1(τ n+1 ,τ n ] decF0 (H) ⊂ 1[0,τ n+1 ] decF0 (H) + 1(τ n+1 ,τ n ] decF n+1 (H). In a similar 1

1

1

1

1

τ1

n ] decF n (H) ⊂ 1 n n+1 decF n (H)+1 n+1 n+1 way we get 1(τin ,τi+1 (τ ,τ ] (τ ,τ τ τ i

i

2i+1

i

2i+1

2(i+1) ]

decF

n+1 τ2i+1

(H)

for every i = 1, 2, . . . , pn − 1. Therefore, for every n ≥ 1 one has decπn (H) ⊂ decπn+1 (H).   Lemma 3.3.7 Let H ⊂ L2 ([0, T ] × , F , Rr ) and (πn )∞ n=1 be a sequence of partitions of the interval [0, T ] of the form πn = {0 < τ1n < τ2n < . . . < τpnn −1 < τpnn } with τin = iδn for i = 1, . . . , pn , where δn = T /2n . Then Lim[decπn (H)] ⊂ decF (co H) and decF (co H) \ Lim[decπn (co H)] = ∅. Proof By virtue of Lemmas 3.3.4 and 3.3.6,for every n ≥ 1 one has decπn (H) ⊂ decπn+1 (co H) ⊂ decF (H). Therefore, cl[ ∞ n=1 decπn (H)] ⊂ decF (H), which by Lemma 1.3.5 of Chapter 1 implies that Lim[decπn (H)] ⊂ decF (H). To see that

3 Decomposable Subsets of Lp (T , F , μ, X)

100

n , τ n ] there decF (co H) \ Lim[decπn (co H)] = ∅, let us note that for every t ∈ (τi−1 i t t n . Therefore, there are elements of dec (H) exists B ∈ F such that B ∈ Ft \Fτi−1 F

that do not belong to decFτ n (H). Then decF (H) \ Lim[decπn (H)] = ∅. i−1

 

3.4 Conditional Expectation of Subsets of Lp (T , F , µ, X) We shall now present properties of conditional expectations of subsets of the space Lp (T , F, μ, X), where X is a Banach space. To begin with let us recall that for a given g ∈ Lp (T , F, μ, X) and a sub-σ -algebra G of F, the conditional expectation of g given G, denoted by E[g|G], is defined to be the (a.s. unique) X-random  variable such that: (1) E[g|G] is G-measurable, and (2) A E[g|G]dP = A gdP for every A ∈ G. The existence and uniqueness of E[g|G] follow from the Radon– Nikodym theorem. Properties of conditional expectations can be found in the classical handbooks on probability theory. Given a sub-σ -algebra G of F and a nonempty set H ⊂ Lp (T , F, μ, X) the set-valued conditional expectation E[H|G] of a set H relative to G is defined to be an G-measurable set-valued mapping E[H|G] : T → Cl(X) such that SG (E[H|G]) = decG {E[h|G] : h ∈ H}, where SG (E[H|G]) denotes the set of all G-measurable selectors to E[H|G]. In particular, if H = S(F ), where S(F ) is subtrajectory integrals of a multifunction F ∈ A(T , X), then E[S(F )|G] is denoted by E[F |G], and said to be the set-valued conditional expectation of F relative to G. Theorem 3.4.1 Given a sub-σ -algebra G of F, and a nonempty set H ⊂ Lp (T , F, μ, X) there exists a (a.s. unique) G-measurable set-valued mapping E[H|G] : T → Cl(X) such that SG (E[H|G]) = decG {E[h|G] : h ∈ H}. Proof Let M = decG {E[h|G] : h ∈ H}. It is clear that M is the closed Gdecomposable subset of the space Lp (T , G, μ, X). Then by Theorem 2.3.3 of Chapter 2, there exists a unique G-measurable set-valued mapping E[H|G] : T → Cl(X) such that SG (E[H|G]) = M.   Corollary 3.4.1 If F ∈ A(T , X), then for every sub-σ -algebra G of F, one has SG (E[F |G]) = cl{E[f |G] : f ∈ S(F )}, where the closure is taken with respect to the norm topology of the space Lp (T , F, μ, X). Proof Let us note that a set {E[f |G] : f ∈ S(F )} is G-decomposable. Indeed, for every u, v ∈ {E[f |G] : f ∈ S(F )}, there exist f, g ∈ S(F ) such that for every A ∈ G one has 1A u + 1A∼ v = 1A E[f |G] + 1A∼ E[g|G] = E[1A f |G] + E[1A∼ g|G] = E[1A f + 1A∼ g|G]. But A ∈ G ⊂ F. Then 1A f + 1A∼ g ∈ S(F ), because S(F ) is F-decomposable. Therefore, 1A u + 1A∼ v ∈ {E[f |G] : f ∈ S(F )} for every u, v ∈ {E[f |G] : f ∈ S(F )} and A ∈ G. By the definition of E[F |G] one has E[F |G] = E[S(F )|G], which implies that SG (E[F |G]) = SG (E[S(F )|G]). But SG (E[S(F )|G]) = decG {E[f |G] : f ∈ S(F )} = cl[decG {E[f |G] : f ∈ S(F )}]. Then SG (E[F |G]) = cl{E[f |G] : f ∈ S(F )}.  

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101

Similarly to proofs of properties of the conditional expectation of set-valued mappings (see [24], Th. 5.2) we obtain properties of conditional expectations of nonempty subsets of the space Lp (T , F, μ, X). We begin with the following result. Lemma 3.4.1 For every nonempty weakly compact set H ⊂ Lp (T , F, μ, X) and a sub-σ -algebra G of F, a set {E[f |G] : f ∈ H} is a weakly compact subset of the space Lp (T , G, μ, X). ∞ Proof Let (un )∞ n=1 be a sequence of the set {E[f |G] : f ∈ H}, and (fn )n=1 be a sequence in H such that un = E[fn |G] for n ≥ 1. By weak compactness of the set H there exists g ∈ H and a subsequence (fnk ) of (fn ) such that fnk  g as k → ∞. Hence, it follows that E[fnk |G]  E[g|G], because E[· |G] is a linear mapping defined on Lp (T , F, μ, X). But, unk = E[fnk |G] and unk  u as k → ∞, where u = E[g|G] ∈ {E[f |G] : f ∈ H}. Then a sequence (un )∞ n=1 possesses a subsequence (unk )∞ k=1 weakly convergent to u ∈ {E[f |G] : f ∈ H}. Thus, a set {E[f |G] : f ∈ H} is a sequentially weakly compact subset of the space ˘ Lp (T , G, μ, X), which by Eberlein–Smulian theorem implies that the set {E[f |G] : f ∈ H} is weakly compact.  

Theorem 3.4.2 Let K and H be nonempty closed bounded subsets of the space Lp (T , F, μ, X), and let G be a sub-σ -algebra of F. Then  (i) T h(E[K|G](t), E[H|G](t))μ(dt) ≤ H (K, H), (ii) E[co|G] = co E[K|G], μ-a.e., (iii) E[ξ · K|G] = ξ · E[K|H], μ-a.e. for ξ ∈ L∞ (T , F, μ, X), (iv) if K, H ⊂ Lp (T , F, μ, X) are closed convex integrably bounded, then E[K + H|G] = E[K|G] + E[H|G], μ-a.e. Proof (i) By the definitions of E[K|G], and E[H|G], Theorem 2.3.4 of Chapter 2, and Remark 2.3.2 of Chapter 2, one has 

¯ h(E[K|G](t), E[H|G](t))μ(dt) = T

 sup{inf{|x − y| : y ∈ E[H|G](t)} : x ∈ E[K|G](t)}}μ(dt) = T





sup inf T





|u(t) − v(t)|μ(dt) : v ∈ SG (E[H|G]) : u ∈ SG (E[K|G]) =

    sup inf |u(t) − v(t)|μ(dt): v ∈ decG {E[g|G] : g ∈ H} : u ∈ decG {E[f |G] : f ∈ K} T







= sup inf



|E[f |G](t) − E[g|G](t)|μ(dt) : g ∈ H : f ∈ K ≤ T

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102









E[|f (t) − g(t)||G]μ(dt) : g ∈ H : f ∈ K ≤

sup inf T









¯ |f (t)−g(t)|μ(dt) : g ∈ H : f ∈ K ≤ h(K, H) ≤ H (K, H).

sup inf T

 ¯ In a similar way, we also get T h(E[H|H](t), E[K|G](t))μ(dt) ≤ H (K, H).  Then T h(E[K|H](t), E[H|G](t))μ(dt) ≤ H (K, H). (ii) By the definition of the set-valued conditional expectation, we have SG (E[co K|G]) = decG (co{E[f |G] : f ∈ K}), which by Theorems 3.3.3 and 2.3.5 of Chapter 2 implies that SG (E[co K|]) = co[decG {E[f |G] : f ∈ K}] = co SG (E[K|G]) = SG (co E[K|G]). Then E[co K|G] = co E[K|G] μ-a.e. (iii) Let ξ ∈ L∞ (T , G, μ, X). Similarly as above we obtain SG (E[ξ · K|G]) = decG {E[ξf |G] : f ∈ K}. But, for every f ∈ K one has E[ξf |G] = ξ E[f |G], and decG {ξ E[f |G] : f ∈ K} = ξ ·decG {E[f |G] : f ∈ K} = ξ ·SG (E[K|G]) = SG (E[ξ ·H|G]). Indeed, it is clear that ξ ·{E[f |G] : f ∈ K} ⊂ ξ ·decG {E[f |G] : f ∈ K}. For every u ∈ decG {E[f |G] : f ∈ K} there exists a sequence (un ) of decG {E[f |G] : f ∈ K} such that | un − u | → 0 as n → ∞, Nn n partition (Ak )N of T , and a with un = k=1 1Ak fk for an G-measurable Nn k=1 Nn Nn family (fk )k=1 ⊂ K. But, un = E[ k=1 1Ak fk |G] = k=1 1Ak E[fk |G]  n 1 {ξ E[f |G] : f ∈ K} ⊂ dec for n ≥ 1. Then ξ un ∈ N G {ξ E[f |G] : k=1 Ak f ∈ K}. Similarly from | ξ un − ξ u | → 0 as n → ∞, it follows that ξ u ∈ decG {ξ E[f |G] : f ∈ K}. Then ξ · decG {E[f |G] : f ∈ K} ⊂ decG {ξ E[f |G] : f ∈ K}. But, immediately from the inclusion ξ · {E[f |G] : f ∈ K} ⊂ ξ ·decG {E[f |G] : f ∈ K} and the G-decomposability of ξ ·decG {E[f |G] : f ∈ K} it follows that decG (ξ · {E[f |G] : f ∈ K}) ⊂ ξ · decG {E[f |G] : f ∈ K}. Thus, decG (ξ · {E[f |G] : f ∈ K}) = ξ · decG {E[f |G] : f ∈ K}. Then SG (E[ξ · K|G]) = SG (ξ · E[K|G]), which by Corollary 2.3.1 of Chapter 2 implies that E[ξ · K|G] = ξ · E[K|G], μ-a.e. (iv) By the definition of the set-valued conditional expectation, and properties of decomposable hulls, we obtain SG (E[(K + H)|G]) = decG {E[(f + g)|G] : f ∈ K, g ∈ H} = cl(decG {E[f |G ] + E[g|G ]:f ∈ K, g ∈ H})=cl[decG ({E[f |G ]:f ∈ K} + {E[g|G ]:g ∈ H})]

= cl{decG {E[f |G] : f ∈ K} + decG {E[g|G] : g ∈ H}} ⊂ cl[decG {E[f |G ] : f ∈ K} + decG {E[g|G ] : g ∈ H}] = cl[SG (E[K|G ]) + SG (E[H|G ])].

By Theorem 2.3.6 of Chapter 2, we get cl[SG (E[K|G]) + SG (E[H|G])] = SG (E[K|G] + E[H|G]). By integrable boundedness of K and H, and by Jensen

3.5 Set-Valued Martingales and Martingale Selectors

103

inequality, it follows that E[K|G] and E[H|G] are integrably bounded. Thus, E[K|G] and E[H|G] are set-valued mappings with convex weakly compact values. Hence, by Corollaries 1.2.6 and 1.2.4 of Chapter 1, it follows that E[K|G]+E[H|G] is the set-valued mapping with convex weakly compact values. Therefore, SG (E[K|G] + E[H|G]) = SG (E[K|G] + E[H|G]). Then SG (E[K + H|G]) ⊂ SG (E[K|G] + E[H|G]), which by Corollary 2.3.1 of Chapter 2 implies that E[K +H|G] ⊂ E[K|G]+E[H|G], μ a.e. Starting now from SG (E[K|G]+E[H|G]) in a similar way we obtain SG (E[K|G] + E[H|G]) ⊂ SG (E[K + H|G]), which implies that E[K|G] + E[H|G] ⊂ E[K + H|G], μ a.e.  

3.5 Set-Valued Martingales and Martingale Selectors Let PF = (, F, F, P ) be a complete filtered probability space with filtration F = (Ft )t≥0 satisfying the usual conditions. An F-non-anticipative set-valued stochastic process F : R+ ×  → Cl(Rd ) with convex values is said to be the set-valued Fmartingale or simply set-valued martingale, if SFt (Ft ) = ∅ and E[SFt (Ft )|Fs ] = Fs a.s., for every 0 ≤ s ≤ t < ∞. By Corollary 3.4.1, the last condition can be written as SFs (Fs ) = cl{E[ft |Fs ] : f ∈ SFt (Ft )} for every 0 ≤ s ≤ t < ∞. It is clear that if F satisfies the above conditions, and is single-valued, then it is an F-martingale. A d-dimensional F-martingale f = (ft )t≥0 is said to be a martingale selector of the set-valued F-martingale F = (Ft )t≥0 , if ft ∈ SFt (Ft ) for every t ≥ 0. In what follows, the set of all martingale selectors of the set-valued F-martingale F = (Ft )t≥0 is denoted by MS(F ). From [76, Prop.2] it follows that if a filtration F is continuous, then every set-valued F-martingale F = (Ft )t≥0 admits a continuous martingale selector. Existence of cádlág selectors for set-valued martingale has been considered by M. Michta in the paper [71]. If F is uniformly square integrably bounded, then MS(F ) ⊂ S(F, Rd ), where S(F, Rd ) is the space of all ddimensional continuous F-martingales x = (xt )t≥0 such that E[supt≥0 |xt |2 ] < ∞. It is clear that S(F, Rd ) is a normed space with the norm | · | defined by | x | 2 = E[supt≥0 |xt |2 ] for x ∈ S(F, Rd ). It can be verified that S(F, Rd ) is a closed subset of the space L2 (, F, X), where X = Cb (R+ , Rd ) is a Banach space of continuous bounded functions x : R+ → Rd with the supremum norm | · |. Then (S(F, Rd ), | · |) is a Banach space. It can be proved that if PF is a separable filtered probability space, then MS(F ) is a separable metric space with a metric ρ defined by the norm | · |. We shall now prove the following results. Lemma 3.5.1 If PF is a separable filtered probability space, then (S(F, Rd ), | · |) is a separable Banach space. Proof Let X = Cb (R+ , Rd ) and let us recall that every d-dimensional continuous process x = (xt )t≥0 can be defined as a (F, β(X))-measurable mapping x :  → X, where β(X) is the Borel σ -algebra on X. Then S(F, Rd ) ⊂ L2 (, F, X), where

3 Decomposable Subsets of Lp (T , F , μ, X)

104

L2 (, F, X) denotes the Banach space of Bochner square integrable functions x :  → X with the norm | · | defined by | x | 2 = E[|x|2 ]. It is clear that L2 (, F, X) is separable, because X is separable. Therefore, to conclude the proof it is enough to verify that S(F, Rd ) is a closed subset of L2 (, F, X). Let (f n )∞ n=1 be a sequence d 2 of S(F, R ) convergent in the norm topology of L (, F, X) to f ∈ L2 (, F, X). Then | f n −f | → 0 as n → ∞, and E[ftn |Fs ] = fsn a.s. for every 0 ≤ s ≤ t < ∞, and n ≥ 1. By Jensen’s inequality, it follows that E[|E[ft |Fs ] − fs |2 ] ≤ 2E[|E[ft |Fs ] − ftn |Fs ]|2 ] + 2E[|E[ftn |Fs ] − fs |2 ] ≤ 2E[E[|ft − ftn |2 |Fs ]] + 2E|fs − fsn |2 ≤ 4 | f n − f |, for every 0 ≤ s ≤ t < ∞ and n ≥ 1, which implies that E[ft |Fs ] = fs a.s. for every 0 ≤ s ≤ t < ∞. Then f ∈ S(F, Rd ). Thus, S(F, Rd ) is a closed subset of the separable Banach space L2 (, F, X).   Lemma 3.5.2 For every uniformly square integrably bounded set-valued martingale F , a set MS(F ) is a closed subset of S(F, Rd ). Proof It is clear if MS(F ) = ∅. Suppose, MS(F ) = ∅ and let (f n )∞ n=1 be a sequence of MS(F ) convergent in the norm | · | to f = (ft )t≥0 ∈ S(F, Rd ). For every t ≥ 0 and n ≥ 1, one has dist(ft , SFt (Ft )) ≤ 2E[|ft − ftn |2 ] + 2dist(ftn , SFt (Ft )) ≤ E[sup |ft − ftn |2 ] ≤ 2 | f − f n |, t≥0

because ftn ∈ SFt (F ). Then dist(ft , SFt (Ft )) = 0 for every t ≥ 0, which implies that ft ∈ SFt (Ft ) for every t ≥ 0. Therefore, f ∈ MS(F ).   From [21, Theorem 3.2], by discretization of the set-valued martingale F at points t = 0, 1, 2, . . ., it can be proved that MS(F ) = ∅, and that SFn (Fn ) = )]}, where F = (Fn )∞ , Pn [(fk )∞ ] = fn for n = 0, 1, 2, . . ., cl{Pn [MS(F n=0 k=1 ∞ and (fk )k=1 ∈ MS(F ). Basing on such approach, the martingale selectors of F can be defined (see [71], Prop. 3, and [76], Prop. 2) by setting ft =  ∞ ∞ k=1 1[k−1,k) (t)E[fk |Ft ] a.s. for t ≥ 0 and every (fk )k=1 ∈ MS(F ). It is clear that every f ∈ MS(F ) can be defined by the above formula. Indeed, ∞ for every k ≥ 1 and t ∈ [k − 1, k), one has f = E[f |F ]. Therefore, f = t k t t k=1 1[k−1,k) (t)ft = ∞ 1 (t)E[f |F ] a.s. for t ≥ 0. k t [k−1,k) k=1 Lemma 3.5.3 Let F = (Ft )t≥0 be a set-valued F-martingale defined on a filtered probability space PF , and let Pt (f ) = ft for t ≥ 0 and f = (ft )t≥0 ∈ MS(F ). Then decFt {Pt [MS(F )]} = SFt (Ft ) for t ≥ 0. = (Fn )∞ and Proof Let F F = (Fn )∞ n=0 n=0 . It is clear that F is a discrete setvalued F-martingale such that for every t ≥ 0, there exists n ≥ 1 such that

3.6 Notes and Remarks

105

n − 1 ≤ t < n and Pt [MS(F )] ⊂ {E[fn ]Ft ] : fn ∈ SFn (Fn )}. Let us observe that {E[fn ]Ft ] : fn ∈ SFn (Fn )} is for fixed t ≥ 0, an Ft -decomposable subset of the space L(, Ft , Rd ). Indeed, let t ≥ 0 be fixed and n ≥ 1 be such that n − 1 ≤ t < n. Then for every u, v ∈ {E[fn ]Ft ] : fn ∈ SFn (Fn )} there exist fn , gn ∈ SFn (Fn ) such that u = E[fn |Ft ] and v = E[gn |Ft ]. Therefore, for any A ∈ Ft one has 1A u + 1A∼ v = E[1A fn + 1A∼ gn |Ft ]. But, SFn (Fn )} is an Fn -decomposable subset of the space L2 (, Fn , Rd ) and A ∈ Ft ⊂ Fn . Then 1A u+1A∼ v ∈ SFn (Fn )} and therefore, 1A u+1A∼ v ∈ {E[fn ]Ft ] : fn ∈ SFn (Fn )}. Now, immediately from the inclusion Pt [MS(F )] ⊂ {E[fn ]Ft ] : fn ∈ SFn (Fn )} and definitions of the set-valued conditional expectation of multifunctions and the set-valued martingales, it follows that dec{Pt [MS(F )]} ⊂ cl{E[fn ]Ft ] : fn ∈ SFn (Fn )} = SFt (E[Fn |Ft ]) = SFt (Ft ). On the other hand, for every n ≥ 1 and t ∈ [n − 1, n), one gets SFt (Ft ) = SFt (E[Fn |Ft ]) = cl{E[f |Ft ] : f ∈ SFn (Fn ) ⊂ cl{Pt (f ) : f ∈ MS(F )} ⊂ dec{Pt (f ) : f ∈ MS(F )} = dec{Pt [MS(F )]}. Therefore, SFt (Ft ) = dec{Pt [MS(F )]}.

 

3.6 Notes and Remarks The content of this chapter is based on N. Dunford and J.T. Schwarz [15], A. Alexiewicz [2], A. Fryszkowski [16], F. Hiai and H. Umegaki [24], Sh. Hu and S.P. Papageorgiou [27], J. Distel and J. Uhl [13], Ch. Hess [21], M. Michta [71] and M. Kisielewicz [38]. In particular, the results of Section 3.1 are based on A. Alexiewicz [2], N. Dunford and J.T. Schwarz [15], and Sh. Hu and S.P. Papageorgiou [27]. The results of Section 3.2 come from A. Fryszkowski [16], F. Hiai and H. Umegaki [24], and J. Distel and J. Uhl [13]. In particular, proofs of Lemmas 3.2.2 and 3.2.3 are constructed similar to the proofs of result dealing with closed convex sets presented in J. Distel and J. Uhl [13]. Similarly to results dealing with convex hulls presented in N. Dunford and J.T. Schwarz [15], the results dealing with decomposable hulls are proved. Some properties of closed decomposable sets are proved by similar approach as results presented in F. Hiai and H. Umegaki [24] and Sh. Hu and S.P. Papageorgiou [27], dealing with properties of measurable bounded set-valued mappings and their subtrajectory integrals. Lemma 3.2.5 is an extension of author’s result (see [38], Th. 2.3 of chap. III) dealing with a Lebesgue measure space ([α, β], L, μ) to the case of the finite measure space (T , F, μ). The result presented in [38] has been extended in [27, Th. 3.17 of Chap. 2] to a weakly closed subsets of the space Lp (T , F, X). Theorem 3.3.5 comes from M. Michta [74], and Theorem 3.3.6 from A. Fryszkowski [17]. In particular, if K ⊂ L(T , F, X) is closed and decomposable, then results of Section 3.4 cover with similar results dealing with conditional expectations of multifunctions presented in [24]. The definition of the

106

3 Decomposable Subsets of Lp (T , F , μ, X)

set-valued conditional expectation of subsets of the space L2 (, F, Rd ) has been presented first in the paper [49]. The results of Section 3.4 are based on results of F. Hiai and H. Umegaki [24] dealing with conditional expectation of set-valued mappings. The results of Section 3.5 are based on M. Michta [71], M. Michta and L. Rybi´nski [76], Ch. Hess [21], and M. Kisielewicz [50]. The proof of Fubini theorem and Theorem 3.1.6 can be found in N. Dunford and J.T. Schwarz [15]. Lemma 3.2.5 was proved first in the author monograph [38]. Theorem 3.3.5 are taken from M. Michta [74] and Theorem 3.3.6 from A. Fryszkowski [17]. The results of Section 3.4 are extensions of some results of F. Hiai and H. Umegaki presented in [24], to the case of conditional expectations of subsets of the space L2 (, F, Rd ).

Chapter 4

Aumann Stochastic Integrals

In this chapter we present the definition and properties of Aumann stochastic integrals of set-valued stochastic processes F : R+ ×  → Cl(Rd ) and subsets of the space Lp (R+ × , β ⊗ F, Rd ). We begin with the definition and properties of the Aumann integrals of subsets of the space Lp (T , F, μ, X), where (X, | · |) is a separable Banach space.

4.1 Aumann Integrals of Subsets of Lp (T , F , µ, X) Let (T , F, μ) be σ -finite measure space and (X, | · |) a separable  Banach space. For a given multifunction F ∈ A(T , X), the Aumann integral A F (t)μ(dt) of F over a set A ∈ F is defined (see [5, 24, 27]) to be a set JA (S(F )) ⊂ X, where S(F ) is a subtrajectory integral of F (see Section 1.3 of Chapter 1), and JA : Lp (T , F,μ, X) → X is a linear continuous mapping defined above, i.e., JA (f ) = A f (t)μ(dt) for every f ∈ Lp (T , F, μ, X). In what follows, a linear mapping JA will be denoted by J (·, A). The above definition can be applied to a nonempty set K ⊂ Lp (T , F, μ, X) and A ∈ F. In such case the  Aumann integral A K μ(dt) is simply denoted by J (K, A). It will be considered for nonempty bounded, decomposable and integrably bounded subsets of the space Lp (T , F, μ, X). Recall, a set K ⊂ Lp (T , F, μ, X) is said to be integrably bounded if there is an m ∈ Lp (T , F, μ, R+ ) such that for every g ∈ K one has |g(t)| ≤ m(t) p for a.e. t ∈ T . It is clear that every integrably bounded set  K ⊂ L (T , F, μ, X) is uniformly integrable, i.e., is such that limc→0 supf ∈K |f |≥c |f (t)|μ(dt) = 0. It can be proved (see [27], Prop. A.2.52) that a set K ⊂ Lp (T , F, μ, X) is uniformly integrable if and only if it is bounded and for every ε > 0 there is a δ > 0 such that supf ∈K A |f (t)|μ(dt) ≤ ε for every A ∈ F such that μ(A) ≤ δ. Similarly as above, the norm on the space Lp (T , F, μ, X) is denoted by | · |. © Springer Nature Switzerland AG 2020 M. Kisielewicz, Set-Valued Stochastic Integrals and Applications, Springer Optimization and Its Applications 157, https://doi.org/10.1007/978-3-030-40329-4_4

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4 Aumann Stochastic Integrals

Lemma 4.1.1 For every A ∈ F and a nonempty set K ⊂ Lp (T , F, μ, X), one has cl[J (K, A)] = cl[J (K, A)], where closures are taken with respect to norms topologies of the space X and Lp (T , F, μ, X), respectively. Proof For every fixed A ∈ F, and a nonempty set K ⊂ Lp (T , F, μ, X), one has cl[J (K, A)] ⊂ cl[J (K, A)]. On the other hand, for every u ∈ cl[J (K, A)] and ε > 0 there is uε ∈ J (K, A) such that |u − uε | ≤ ε/2. By the definition of J (K, A) there is xε ∈ K such that uε = J (xε , A). Let zε ∈ K be such that | xε − zε | ≤ ε/(2M) with M > 0 such that |J (xε , A) − J (zε , A)| ≤ M | xε − zε |. Then for every u ∈ cl[J (K, A)] and ε > 0 there is zε ∈ K such that |u − J (zε )| ≤ |u − uε | + |uε − J (zε )| ≤ ε/2 + ε/2 = ε, which implies that u ∈ cl[J (K, A)]. Thus, cl[J (K, A)] ⊂ cl[J (K, A)]. Therefore, cl[J (K, A)] = cl[J (K, A)].   Theorem 4.1.1 For every A ∈ F, and a nonempty set K ⊂ Lp (T , F, μ, X), one has cl[J (co K, A)] = cl[co J (K, A)], and cl[J (dec K, A)] = cl[J (dec(K), A)], respectively. Proof By virtue of Corollary 1.2.2 of Chapter 1, and Lemma 4.1.1, one gets cl[J (co K, A)] = cl{J ([co K], A)} = cl{J (co K, A)}. On the other hand, we have J (co(K), A) = co[J (K, A)]. Then cl[J (co K, A)] = cl[co J (K, A)]. In a similar way the equality cl[J (dec(K), A)] = cl[J (dec(K), A)] can be obtained.   Lemma 4.1.2 For every A ∈ F and nonempty sets K, H ⊂ Lp (T , F, μ, X), one has J (α · K + β · H, A) = α · J (K, A) + β · J (H, A) for a every α, β ∈ R. Proof Let A ∈ F, K, H ⊂ Lp (T , F, μ, X) and α, β ∈ R be given. For every u ∈ J (α·K+β ·H, A) there are x ∈ K and y ∈ H such that u = J (α·x +β ·y, A) = α · J (x, A) + β · J (y, A) ∈ α · J (K) + β · J (H, A). Then J (α · K + β · H, A) ⊂ α · J (K, A) + β · J (H, A). Similarly, for every u ∈ α · J (K, A) + β · J (H, A) there are x ∈ K and y ∈ H such that u = α · J (x, A) + β · J (y) = J (α · x + β · y, A) ∈ J (α · K + β · H, A). Then α · J (K, A) + β · J (H, A) ⊂ J (α · K + β · H, A).   Corollary 4.1.1 For every A ∈ F and nonempty sets K, H ⊂ Lp (T , F, μ, X), one has cl[J (K + H, A)] = cl[J (K, A) + J (H, A)]. Proof By virtue of Lemma 4.1.1, one has cl[J (K + H, A)] = cl[J (K + H, A)]. Hence, by Lemma 4.1.2, it follows that cl[J (K + H, A)] = cl[J (K, A)+J (H, A)].   Lemma 4.1.3 Let (X, | · |) be a reflexive Banach space. If H is a nonempty integrably bounded subset of Lp (T , F, μ, X) and f ∈ Lp (T , F, μ, X) is such that J (f, A) ∈ J (H, A) for every A ∈ F of the finite measure μ(A), then f ∈ dec[co H]. Proof Let f ∈ Lp (T , F, μ, X) be such that J (f, A) ∈ J (H, A) for every A ∈ F. ε of T such that We can select for every ε > 0 an F-measurable partition (Aεn )nn=1 ε n+2 p |J (m, An )| < ε/2 for every n = 1, . . . , nε , where m ∈ L (T , F, μ, R+ ) is such that |h(t)| ≤ m(t) for every h ∈ H and a.e. t ∈ T . For every n = 1, . . . , nε , we have J (f, Aεn ) ∈ J (H, Aεn ). Therefore, for every n = 1, . . . , nε ,

4.1 Aumann Integrals of Subsets of Lp (T , F , μ, X)

109

there existsfnε ∈ H such that J (f, Aεn ) = J (fnε , Aεn ) for every n = 1, . . . , nε . ε Let f ε = nn=1 1Aεn fnε . By Lemma 3.3.1 of Chapter 3, we have f ε ∈ dec(H) ⊂ dec{co(H)}. By Corollary 3.3.2 of Chapter 3, one has dec{co(H)} = dec{co(H)}. By Theorems 3.3.3 and 3.3.5 of Chapter 3, dec{co(H)} is a closed convex bounded subset of the space Lp (T , F, μ, X). Therefore, by Theorem 1.2.3 of Chapter 1, it follows that for p > 1 a set dec{co(H)} is weakly compact. By Remark 3.1.4 of Chapter 3, the set dec{co(H)} is also weakly compact for p = 1. Let (εk )∞ k=1 be a sequence of positive numbers εk > 0 convergent to zero, and let g k = f εk for k ≥ 1. We have g k ∈ dec{co(H)} for every k ≥ 1. Therefore, there is a k subsequence (g kj )∞ j =1 of the sequence (g ), weakly convergent to g ∈ dec{co(H)}. Hence, by Dunford–Schwartz theorem it follows that a sequence {J (g kj , A)}∞ j =1 weakly converges to J (g, A) for every A ∈ F. For every j ≥ 1, one has η g kj = f ηj , J (g kj , Anj ) and J (f ηj , A) = J (g kj , A), where ηj = εkj . Furthermore, nηj η η |J (f, A ∩ Anj ) − J (f ηj , A ∩ Anj )|. But |J (f, A) − J (f ηj , A)| ≤ n=1 Nj  n=1

η

η

|J (f, A ∩ Anj ) − J (f ηj , A ∩ Anj )| ≤ 2

Nj  n=1

η

J (m, Anj ) ≤ 2

∞  n=1

ε = ε, 2n+2

where Nj = nηj . Therefore, J (f ηj , A)  J (f, A) for every A ∈ F as j → ∞. On the other hand, for every A ∈ F one has J (f ηj , A)  J (g, A) as j → ∞, because J (f ηj , A) = J (g kj , A). Then J (f, A) = J (g, A) for every A ∈ F, which by [13, Lemma 6 of Chap. 1] implies that f (t) = g(t) for a.e. t ∈ T . Thus, f ∈ dec{co(H)}, because g ∈ dec{co(H)}. In a similar way the case p > 1 can be considered.   Theorem 4.1.2 Let (X, | · |) be a reflexive Banach space. If K and H are nonempty integrably bounded subsets of Lp (T , F, μ, X) such that J (K, A) = J (H, A) for every A ∈ A of the finite measure μ(A), then dec[co K] = dec[co H]. Proof For every f ∈ K and every A ∈ F we have J (f, A) ∈ J (H, A), which by Lemma 4.1.3, implies that f ∈ dec{co(H)}. Then K ⊂ dec{co(H)}. By Theorem 3.3.3 of Chapter 3, a set dec{co(H)} is a closed convex decomposable subset of Lp (T , F, μ, X). Therefore, dec{co(K)} ⊂ dec{co(H)}. In a similar way we also get dec{co(H)} ⊂ dec{co(K)}.   Corollary 4.1.2 Let (X, | · |) be a reflexive Banach space. For every nonempty convex integrably bounded sets K, H ⊂ Lp (T , F, μ, X) such that J (K, A) = J (H, A) for every A ∈ F of the finite measure, one has dec(K) = dec(H).   Corollary 4.1.3 Let (X, | · |) be a reflexive Banach space. If (T , F, μ) is an σ finite measure space, then for every nonempty decomposable integrably bounded sets K, H ⊂ Lp (T , F, μ, X) such that J (K, A) = J (H, A) for every A ∈ F of finite measure, one has co(K) = co(H).

110

4 Aumann Stochastic Integrals

Proof By Theorem 4.1.2, we have dec{co(K)} = dec{co(H)}. Hence, by Theorem 3.3.3 of Chapter 3, it follows that co{dec(K)} = co{dec(H)}, which by the decomposability of sets K and H implies co(K) = co(H).   Corollary 4.1.4 Let (X, | · |) be a reflexive Banach space. If (T , F, μ) is an σ -finite measure space, then for every nonempty decomposable integrably bounded convex sets K, H ⊂ Lp (T , F, μ, X) such that J (K, A) = J (H, A) for every A ∈ A of the finite measure, one has cl(K) = cl(H). Proof The result follows immediately from Corollary 4.1.3 and Corollary 1.2.2 of Chapter 1, because co(K) = cl[co(K)] = cl(K) and co(H) = cl[co(H)] = cl(H).   Theorem 4.1.3 Let (X, | · |) be a reflexive Banach space. If (T , F, μ) is an σ finite non-atomic measure space, then for every nonempty decomposable bounded set H ⊂ Lp (T , F, μ, X), one has cl[J (K, A)] = cl[J (co K, A)] for every A ∈ F. Proof It is clear that cl[J (K, A)] ⊂ cl[J (co K, A)] for every A ∈ F. By Theorem 3.2.1 of Chapter 3, a set cl[J ((K, A)] is a closed convex subset of X. Then for every A ∈ F, one has co[J (K, A)] ⊂ cl[J (K, A)]. But co[J (K, A)] = cl[co J (K, A)] = cl[J (co K, A)]. Then cl[J (co K, A)] ⊂ cl[J (K, A)].   Theorem 4.1.4 Let (X, | · |) be a reflexive Banach space. If (T , F, μ) is a σ finite measure space, then for every nonempty decomposable bounded set H ⊂ Lp (T , F, μ, X) with p ≥ 1, the set J (clw (H), A) is a convex weakly compact subset of X for every A ∈ F. Proof By Remark 3.2.3 of Chapter 3, clw (H) is a convex subset of Lp (T , F, μ, X), which by the linearity of J implies that J (clw (H), A) is a convex subset of X. On the other hand, by Corollary 1.2.6 of Chapter 1, clw (H) is a weakly compact subset of Lp (T , F, μ, X), which by Dunford–Schwarz theorem implies that J (clw (H), A) is weakly compact. Then J (clw (H), A) is a convex weakly compact subset of X.   From Theorem 3.3.2 and Remark 3.3.1 of Chapter 3, we obtain the following result. Theorem 4.1.5 For every decomposable sets K, H ∈ Cl(Lp (T , F, μ, X)), one has H¯ p (K, H) = T supn≥1 infm≥1 |f n (t) − g m (t)|p μ(dt), where (f n )∞ n=1 and (g m )∞ are representation Castaing of K and H, respectively.   m=1 Corollary 4.1.5 A decomposable set K ⊂ Lp (T , F, μ, X) is integrably bounded if and only if it is a bounded subset of Lp (T , F, μ, X). Proof If there exists a function m ∈ Lp (T , F, μ, R+ ) such that |h(t)| ≤ m(t) for every h ∈ K and a.e. t ∈ T , then H¯ (K, {0}) < ∞. If a decomposable set p K  every h ∈nK and every A ∈ F one has  ⊂ L (T , F, μ, X) is bounded, then for ¯ |h|dτ ≤ sup{J (|h|, A) : h ∈ K} = A A supn≥1 |f (t)|μ(dt) = H (K, {0}) < ∞,

4.1 Aumann Integrals of Subsets of Lp (T , F , μ, X)

111

where (f n )∞ n=1 is a representation Castaing of K. Hence, by [13, Lemma 6 of Chap. 1], one has |h(t)| ≤ supn≥1 |f n (t)| for a.e. t ∈ T .   Theorem 4.1.6 For every nonempty set K ⊂ Lp (T , F, μ, X) and  decomposable ∗ ∗ n A ∈ F one has s(x , J (K, A)) = A supn≥1 x , f (t) μ(dt) for every x ∗ ∈ X∗ , where (f n )∞ n=1 is a representation Castaing of K. Proof By the definition of the support function s(·, J (K, A)), it follows that s(x ∗ , J (K, A)) = sup{ x ∗ , u : u ∈ J (K, A)}. But for every u ∈ J (K, A) there exists h ∈ K such that x ∗ , u = A x ∗ , h(t) μ(dt). Hence, by Remark 3.3.1 of Chapter 3, and the equality K = dec{fn : n ≥  1}, it follows ! that ∗, s(x ∗ , J (K, A)) = sup{ x ∗ , u : u ∈ J (K, A)} = sup{ x h(t)μ(dt) : h ∈ A  K} = suph∈K A x ∗ , h(t) μ(dt) = A supn≥1 x ∗ , fn (t) μ(dt).   Remark 4.1.1 If (X, | · |) is a separable Banach space, then immediately from Theorem 2.3.3 of Chapter 2, it follows that the Aumann integral of a closed decomposable set of H ⊂ Lp (T , F, μ, X) covers with the set J (S(F ), A), where S(F ) is subtrajectory integral of the measurable multifunction F : T → Cl(X) such that S(F ) = H. In what follows, the set J (S(F ), A) is denoted by A F (t)μ(dt) and said to be the Aumann integral ofa multifunction F . It can be defined immediately  by the formula A F (t)μ(dt) = { A f (t)μ(dt) : f ∈ S(F )}.    We shall now present the basic properties of Aumann integral A F (t)μ(dt). Theorem 4.1.7 Let (T , F, μ) be a σ -finite measurable space, and let (X, | · |) be a separable Banach space. If F, G ∈ A(T , X), then for every A ∈ F, one has   (i) cl A co F (t)μ(dt) = co A F (t)μ(dt),    (t) + G(t)]μ(dt) = cl[ A F (t)μ(dt) + A G(t)μ(dt)], (ii) cl A ([F   (iii) s(x ∗ , A F (t)μ(dt)) = A s(x ∗ , F (t))μ(dt) for every x ∗ ∈ X∗ ,  (iv) if (T , F, μ) is non-atomic, then cl A F (t)μ(dt) = cl A co F (t)μ(dt), (v) if (T , F, μ) is non-atomic and X is finite dimensional, then A F (t)μ(dt)  = A co F (t)μ(dt),  (vi) A h(co  F (t)μ(dt), coG(t)μ(dt)) ≤ Ah(F (t), G(t))μ(dt), (vii) h(cl A F (t)μ(dt), cl A G(t)μ(dt)) ≤ A h(F (t), G(t))μ(dt).  Proof (i) By the definition  of the Aumann integral A F (t)μ(dt)  and Theorem 4.1.1, one gets cl A co F (t)μ(dt) = co[J (S(F ), A)] = co A F (t)μ(dt). (ii) By the  definition of the Aumann integral, and Theorem 2.3.6 of Chapter 2, one gets A ([F (t) + G(t)]μ(dt) = cl[J (S(F + G), A)] = cl[J (S(F ) + S(G), A)].  Hence, by Corollary 4.1.1 it follows cl A [F (t) + G(t)]μ(dt = cl[J (S(F ) +   J (S(G)] = cl[ A F (t)μ(dt) + A G(t)μ(dt)]. (iii) The result follows from Theorem 4.1.6, because by Corollary 2.3.2 of Chapter 2, there is a sequence (fn )∞ n=1 ⊂ S(F ) such that F (t) = cl{fn (t) : n ≥ 1} for t ∈ T and S(F ) = dec{fn : n ≥ 1}. Therefore, by the definition of A F (t)μ(dt) and Theorem 4.1.6, one

112

(iv)

(v) (vi) (vii)

4 Aumann Stochastic Integrals

  has s(x ∗ , A F (t)μ(dt)) = s(x ∗ , J (S(F ), A)) = A sup{ x ∗ , x : x ∈  S(F )}μ(dt) = A s(x ∗ , F (t))μ(dt) for every x ∗ ∈ X∗ .  The result follows from (i), because cl A co F (t)μ(dt) = co A F (t)μ(dt), and by virtue Theorem 3.2.1 of Chapter 3, the set  cl[J (S(F ), A)] is convex. But, cl A F (t)μ(dt) = cl[J (S(F ), A)] and cl Aco F (t)μ(dt) = cl[J  (S(co F ), A)]. Then by Theorem 4.1.3, we get cl A F (t)μ(dt) = cl A co F (t)μ(dt). The result follows immediately from (iv), and Theorem 3.2.1 of Chapter 3. The result follows immediately from (iii) of Lemma 1.3.3 of Chapter 1. It is clear that



   F (t) μ(dt), G(t) μ(dt) . h¯ cl F (t) μ(dt), cl G(t) μ(dt) = h¯ A

A

A

A

By the definition of the Hausdorff metric, and Theorem 2.3.4 and Remark 2.3.2 of Chapter 2, it follows that

  h¯ F (t) μ(dt), G(t) μ(dt) 

A

A

  |f (t) − g(t)|μ(dt) : f ∈ S(G) : f ∈ S(F )



= sup inf A





¯ (t), G(t))μ(dt) h(F

sup{inf{|x − y| : y ∈ G(t)} : x ∈ F (t)} =

= A

A

 ≤

h(F (t), G(t))μ(dt). A

In a similar way, we obtain

   ¯h cl G(t) μ(dt), cl F (t) μ(dt) ≤ h(F (t), G(t))μ(dt). A

Then h(cl



A



A F (t), μ(dt), cl A G(t) μ(dt))

A

≤



A h(F (t), G(t))μ(dt).

 

Remark 4.1.2 The result (i) of Theorem 4.1.7 is satisfied (see [27], Th. 5.14 of Chap. 2, [25], Th. 4, Part I, [60], Th. 1.9 of Chap. 18), if F : T → P(Rd ) is graph measurable and F (t) ⊂ Rd+ for every t ∈ T .   Corollary 4.1.6 If (T , F, μ) is a σ -finite non-atomic measure space and F : T →  Cl(Rd ) is measurable and integrably bounded, then A F (t) μ(dt) is a compact convex subset of the space Rd .  Proof By (v) of Theorem 4.1.7 it follows that A F (t)μ(dt) is convex. By inte grable boundedness of F and (vii) of Theorem 4.1.7, it follows that A F (t)μ(dt) is a bounded subset of Rd . By Corollary 2.3.4 of Chapter 2, and Theorem 1.2.3

4.1 Aumann Integrals of Subsets of Lp (T , F , μ, X)

113

of Chapter 1, subtrajectory integrals S(co F ) is a weakly compact subset of the space Lp (T , F, Rd ). Then J (S(co F ), A) = A co F (t)μ(dt) is a closed subset of Rd , because J (S(co F ), A) is weakly compact. Hence, by (v) of  Theorem 4.1.7, it follows that A F (t)μ(dt) is also closed subset of Rd . Thus, A F (t)μ(dt) is a compact convex subset of Rd .   Lemma 4.1.4 If (T , F, μ) is a finite measure space, (X, | · |) is a Banach space and f : E → X is Bochnerintegrable with respect to μ, then for every E ∈ F with the finite measure, one has E f (t) μ(dt) ∈ μ(E) · co f (E).  Proof Suppose there exists a set E ∈ F with μ(E) > 0 such that E f (t) μ(dt) ∈ μ(E) · co f (E). By Theorem 1.2.2 of Chapter 1, there exists x ∗ ∈ X∗ such that x

∗





f (t)μ(dt) < inf{x ∗ (u) : u ∈ μ(E) · cof (E)} E

≤ inf{x ∗ (u) : u ∈ μ(E) · f (E)} ≤ μ(E)x ∗ [f (t)]  for every t ∈ E. Then for every t ∈ E one gets [μ(E)]−1 · x ∗[ E f (t)μ(dt)] < x ∗ [f (t)]. Hence, by integrating over E it follows that x ∗ [ E f (t)μ(dt)] < ∗ ∗ E x ∗ [f (t)]μ(dt). But, by Theorem 1.4.1  of Chapter 1, one has x [ E f (t)μ(dt)] =   E x [f (t)]μ(dt). Contradiction. Then E f (t) μ(dt) ∈ μ(E) · co f (E). Lemma 4.1.5 Let (T , F, μ) be a finite measure space and (X,  |·|) a Banach space. Then for every closed convex set A ⊂ X and E ∈ F one has E A μ(dt) = μ(E)·A.  Proof It is obvious that μ(E)   · A ⊂ E A μ(dt), becausefor every a ∈ A one has μ(E) · a = E a μ(dt)  ∈ { E f (t) μ(dt) : f ∈ S(A)} = E A μ(dt). On the other function f : E → X hand, for every z ∈ E A μ(dt) there exists a measurable  such that f (t) ∈ A for a.e. t ∈ E, and such that z = E f (t) μ(dt). By virtue of Lemma 4.1.4,  there exists ξ ∈ co f (E) ⊂ A such that z = μ(E) · ξ . Therefore, for every z ∈ E μ(dt) one has z ∈ μ(E) · A. Then E A μ(dt) ⊂ μ(E) · A.   We shall now consider some special problems for Aumann integrals of multifunctions F : I → Comp(Rd ), where I is a closed interval of the real line. Assume F : I → Comp(Rd ) is such that for each t ∈ I the function h(F (·), F (t)) is Lebesgue  t+η integrable on I . A point t ∈ I , for which limη→0 η−1 t h(F (τ ), F (t))μ(dτ ) = 0, where μ is a Lebesgue measure on the real line, is called a Lebesgue point of F . A set-valued mapping F : I → Comp(Rd ) is said to be approximately continuous at t ∈ I , if there exists a measurable set B ⊂ I for which t is a point of the density, i.e., such that limδ→0 (1/(2δ)μ([t − δ, t + δ] ∩ B) = 1, and such that the restriction of F to B is continuous at t. From Lusin-Pli´s theorem (see [38], Th. 3.6 of Chap. II), it follows that if F : I → Comp(Rd ) is measurable, then it is approximately continuous a.e. on I . Indeed, let ε > 0 be given. By Lusin-Pli´s theorem, there exists a closed set B ⊂ I with μ(I \ B) < ε and such that the restriction of F to B is continuous. Let D ⊂ B be the set of points of the density of B. Then μ(D) = μ(B) > μ(I ) − ε. If t ∈ D, then F is approximately continuous

114

4 Aumann Stochastic Integrals

at t. Thus the set H ⊂ I of all points of approximate continuity of F has the inner measure greater than μ(I ) − ε. Since ε > 0 is arbitrary, then the inner measure of H is greater than or equal to μ(I ). But H ⊂ I , then its outer measure is less than or equal to μ(I ) showing that H is measurable with measure μ(I ). We prove now the following results. Theorem 4.1.8 If F : I → Comp(Rd ) is measurable and integrably bounded, then almost every t ∈ I is the Lebesgue point of F . Proof Let m ∈ L(I, R) be such that F (t) ≤ m(t) for a.e. t ∈ I . By Lusin-Pli´s theorem and integrable boundedness of F it follows that the function h(F (·), F (t)) is Lebesgue integrable on I . Clearly, we can assume that almost all points of I are at once points of the approximate continuity of F and Lebesgue points of m. Let t ∈ I be such point, and let B ⊂ I be a measurable set for which t is a point of the density such that the restriction of F to B is continuous at t. For η > 0 put B1 (η) = [t, t + η] ∩ B and B2 (η) = [t, t + η] ∩ (I \ B). Thus, given ε > 0 one may choose η = η(ε, t) > 0 sufficiently small such that the following conditions are satisfied:(i) h(F (τ ), F (t)) < 1/6 ε for τ ∈ B1 (η), (ii) μ(B2 (η)) < 1/6 ε η m(t), t+η and (iii) t |m(τ ) − m(t)|μ(dτ ) < 1/3 η ε. Now, we get  t+η  h(F (τ ), F (t))μ(dτ ) = η−1 h(F (τ ), F (t))μ(dτ ) η−1 +η−1



B1 (η)

t

h(F (τ ), F (t))μ(dτ ) < B2 (η)


0 we have η co F (t) = η−1 t co F (t)μ(dτ ). Therefore, for fixed t ∈ I and η > 0 we get  t+η

 h η−1

t+η

t

=η

−1

co F (τ )μ(dτ ), co F (t)





t+η

t+η

co F (τ )μ(dτ ),

h t

t

co F (t)μ(dτ ) .

4.1 Aumann Integrals of Subsets of Lp (T , F , μ, X)

115

By Corollary 4.1.6, and Corollary 1.3.2 of Chapter 1, and (iii) of Theorem 4.1.7, one gets

 t+η h η−1 co F (τ )μ(dτ ), co F (t) t

   = η−1 sup s p,    −1 = η sup  ≤ η−1



t+η t t+η

   co F (t)μ(dτ )  : |p| = 1

 co F (τ )μ(dτ ) − s(p,

t+η t

t+η t

 s(p, co F (τ ))μ(dτ ) −

t+η

t

    s(p, co F (t))μ(dτ ) : |p| = 1

sup{|s(p, co F (τ )) − s(p, co F (t))| : |p| = 1}μ(dτ )

t

≤ η−1



t+η

h(co F (τ ), co F (t))μ(dτ ) ≤ η−1

t



t+η

h(F (τ ), F (t))μ(dτ ), t

because by (iii) of Lemma 1.3.3 of Chapter 1, one has h(co A, co B) ≤ h(A, B)  t+η for every A, B ∈ Cl(Rn ). Hence, it follows limη→0 h(η−1 t co F (τ )μ(dτ ), co F (t)) = 0.   We shall now present some theorems, dealing with Aumann integrals depending on random parameters. To begin with, let us assume that we are given a complete filtered probability space PF = (, F, F, P ) with a filtration F = (Ft )t≥0 satisfying the usual conditions. Let us note that stochastic processes and setvalued stochastic processes can be defined as mappings f : R+ ×  → Rd and F : R+ ×  → Cl(Rd ) such that sections f (t, ·) and F (t, ·) are random and setvalued random variables for every t ≥ 0. These random variables are also denoted by ft and Ft , respectively. Therefore, their mean values can be denoted by E[f (t, ·)] and E[F (t, ·)] or E[ft ] and E[Ft ], respectively. In particular, if ϕ : Cl(Rd ) → R is a continuous function, then a random variable   ω → ϕ(F (t, ω)) ∈ R is denoted by ϕ(F (t, ·)). Similarly a random variable   ω → ϕ(Ft (ω)) ∈ R is denoted by ϕ(Ft ). Theorem 4.1.10 Let F : [0, T ] ×  → Cl(Rd ) be a measurable (F-nonanticipative) convex-valued bounded set-valued stochastic process. Then there exists a sequence (Fn )∞ n=1 of continuous (continuous and F-adapted) set-valued stochastic processes Fn : [0, T ] ×  → Cl(Rd ) with convex values such that Fn (t, ω) ≤ M for a.e. (t, ω) ∈ [0, T ]×, and such that h(Fn (τ, ω), F (τ, ω)) → 0 for fixed ω ∈  and a.e. t ∈ [0, T ], where Fn (t, ω) = sup{|u| : u ∈ Fn (t, ω)} and M > 0 is such that F (t, ω) ≤ M for a.e. (t, ω) ∈ [0, T ] × . Proof Define for every n ≥ 1, the set-valued mapping Fn : [0, T ] ×  → Cl(Rd ) by setting  t F (τ, ω)dτ. Fn (t, ω) = n (t−1/n)∨0

116

4 Aumann Stochastic Integrals

It is clear that Fn (t, ω) is a compact convex subset of Rd . By (vii) of Theorem 4.1.7, t t one gets Fn (t, ω) ≤ n (1−1/n)∨0 F (τ, ω)dτ ≤ n (1−1/n)∨0 Mdτ = M for a.e. (t, ω) ∈ [0, T ] × . Let us note that set-valued mappings Fn (·, ω) and Fn (t, ·) are continuous for fixed ω ∈  and F-measurable for fixed t ∈ [0, T ], respectively. Indeed, by (iii) of Theorem 4.1.7, for every (t, ω) ∈ [0, T ] ×  and n ≥ 1, one has

 s(p, Fn (t, ω)) = n s p,



t

 =n

F (τ, ω)dτ

t

s(p, F (τ, ω))dτ.

(t−1/n)∨0

(t−1/n)∨0

By properties of a mapping F and [38, Th. 3.8 of Chap. II], the support function [0, t] ×   (τ, ω) → s(p, F (τ, ω)) ∈ R is bounded and β([0, t]) ⊗ Ffor every p ∈ Rd . Therefore, a function [0, T ] ×   (t, ω) → measurable t (t−1/n)∨0 s(p, F (τ, ω))dτ ∈ R is continuous in t ∈ [0, T ] for fixed ω ∈ , and F-measurable for fixed t ∈ [0, T ]. To verify that h(Fn (τ, ω), F (τ, ω)) → 0 for a fixed ω ∈  and a.e. t ∈ [0, T ], let us note that by Corollary 4.1.6, Fn (τ, ω) is a compact convex subset of Rd . Then by Corollary 1.3.2 of Chapter 1, and (iii) of Theorem 4.1.7, we get h(Fn (τ, ω), F (τ, ω)) = sup{|s(p, Fn (τ, ω)) − s(p, F (τ, ω))| : |p| = 1}    t

 t    = n sup s p, F (τ, ω)dτ − s p, F (t, ω)dτ  : |p|=1 (t−1/n)∨0 (t−1/n)∨0     t  t   = n sup  s(p, F (τ, ω))dτ − s(p, F (t, ω))dτ ) : |p|=1  ≤n  =n

(t−1/n)∨0

t

(t−1/n)∨0

sup{|s(p, F (τ, ω)) − s(p, F (t, ω))| : |p| = 1}dτ

(t−1/n)∨0 t (t−1/n)∨0

h(F (τ, ω), F (t, ω))dτ = η

−1



t+η

h(F (τ, ω), F (t, ω))dτ t

for every (t, ω) ∈ [0, T ] ×  and n ≥ 1, where η = −1/n. Hence, by Theorem 4.1.9, it follows that h(Fn (τ, ω), F (τ, ω)) → 0 for fixed ω ∈  and a.e. t ∈ [0, T ]. In a similar way we infer that the set-valued stochastic process Fn is F-adapted if F is assumed to be F-non-anticipative.   Theorem 4.1.11 For every measurable (F-non-anticipative) integrably bounded set-valued process F : R+ ×  → Cl(Rd ) there exists a sequence (FN )∞ N =1 of measurable (F-non-anticipative) bounded set-valued processes FN : R+ ×  → T Cl(Rd ) such that E 0 h(F (t, ·), FN (t, ·))dt → 0 for every T > 0 as N → ∞. Proof Let ξN : R+ × → R+ be defined for every N ≥ 1 by setting: ξN (t, ω) = 1 if F (t, ω) ≤ N , and ξN (t, ω) = 0 if F (t, ω) > N for (t, ω) ∈ R+ × , and let FN (t, ω) = ξN (t, ω) · F (t, ω) for (t, ω) ∈ R+ × . It is clear that such defined set-valued stochastic processes FN : R+ ×  → Cl(Rd ) are measurable (F-non-

4.2 Aumann Stochastic Integrals

117

anticipative) and bounded for every N ≥ 1. Furthermore, for every T > 0 and N ≥ 1 one has

E

T 0

T

h(F (t, ·), FN (t, ·))dt = T 1AN h(F (t, ·), FN (t, ·))dt + E 0 1A∼N h(F (t, ·), FN (t, ·))dt, E

0

where AN = {(t, ω) ∈ [0, T ] ×  : F (t, ω) ≤ N } and A∼ N is the T complement of AN . It is clear the E 0 1AN h(F (t, ·), FN (t, ·))dt = 0 and T T E 0 1A∼N h(F (t, ·), FN (t, ·))dt → 0 as N → ∞. Then E 0 h(F (t, ·), FN (t, ·)) dt → 0 as N → ∞.   Corollary 4.1.7 If F : R+ ×  → Cl(Rd ) is a measurable (F-non-anticipative) convex-valued integrably bounded set-valued stochastic process, then for every T > 0 and ε > 0 there exists a sequence (Fnε )∞ n=1 of continuous (continuous and Fadapted) bounded set-valued stochastic processes Fnε : [0, T ] ×  → Cl(Rd ) with T compact convex values such that limn→∞ E 0 h(Fnε (t, ·), F (t, ·)) ≤ ε. Proof By Theorem 4.1.11, there exists a sequence (Fj )∞ j =1 of measurable (F-non+ anticipative) bounded set-valued processes Fj : R ×  → Cl(Rd ) such that T E 0 h(Fj (t, ·), F (t, ·))dt → 0 for every T > 0 as j → ∞. Then for every ε > 0 T there is jε ≥ 1 such that E 0 h(Fjε (t, ·), F (t, ·))dt ≤ ε. By Theorem 4.1.10, there exists a sequence (Fnε )∞ n=1 of continuous (continuous and F-adapted) set-valued stochastic processes Fnε : [0, T ] ×  → Cl(Rd ) with compact convex values such that Fnε (t, ω) ≤ jε for every n ≥ 1 and a.e. (t, ω) ∈ [0, T ] × , and such that limn→∞ h(Fnε (t, ω), Fjε (t, ω)) = 0 for fixed ω ∈  and a.e. t ∈ [0, T ]. Therefore, T limn→∞ E 0 h(Fnε (t, ·), Fjε (t, ·))dt = 0. But for every n ≥ 1 one has T

h(Fnε (t, ·), F (t, ·))dt ≤ T T E 0 h(Fnε (t, ·), Fjε (t, ·))dt + E 0 h(Fjε (t, ·), F (t, ·))dt. E

Then limn→∞ E

T 0

0

h(Fnε (t, ·), F (t, ·)) ≤ ε.

 

4.2 Aumann Stochastic Integrals Let PF = (, F, F, P ) be a complete filtered probability space with a filtration F = (Ft )t≥0 satisfying the usual conditions and let β+ be the Borel σ -algebra on R+ . In what follows, the Lebesgue measure on the real line will  be denoted by dt. For f ∈ Lp (R+ × , β+ ⊗ F, dt × P , Rd ), its integral R+ × f (t, ω)dtdP  will be denoted by E R+ f (t, ·)dtdP . By the Fubini’s theorem, for a.e. t ∈ R+ the function f (t, ·) :  → Rd belongs to the space Lp (, F, Rd ). From

118

4 Aumann Stochastic Integrals

 p d Theorem  3,  it follows that R+ f (t, ·)dt belongs to L (, F, R )  3.1.6 of Chapter and E R+ f (t, ·)dt = R+ × f (t, ω)dtdP . Therefore, for every A ∈ β+ we p can define on the space L (R+ × , β+ ⊗ F, dt × P , Rd ) the linear mapping J (·, A) with values in Lp (, F, Rd ) by setting J (f, A)(ω) = A f (t, ω)dt for every f ∈ Lp (R+ × , β+ ⊗ F, dt × P , Rd ) and ω ∈ . In what follows, J (·, A) will be considered with sets A =: [s, t] for fixed 0 ≤ s < t < ∞, and denoted by Js,t . For simplicity, the space Lp (R+ × , β+ ⊗ F, dt × P , Rd ) will be denoted by Lp (R+ × , β+ ⊗ F, Rd ). Elements of Lp (R+ × , β+ ⊗ F, Rd ) can be treated as p-integrable stochastic processes. Furthermore, we shall consider a subspace Lp (R+ × , F , Rd ) of Lp (R+ × , β+ ⊗ F, Rd ) (see Remark 3.1.2 of Chapter 3) with the σ -algebra F ⊂ β+ ⊗ F of all F-non-anticipative subsets of R+ × . Given a measurable square p-integrably bounded set-valued process  : R+ ×  → Cl(Rd ), by S() the set of all β+ ⊗ F-measurable selectors of  is denoted. If  is F-non-anticipative, i.e., if it is F -measurable, then the set of all F-non-anticipative selectors of  is denoted by SF (). It is clear that by the above assumptions, sets S() and SF () are nonempty closed decomposable subsets of spaces Lp (R+ × , β+ ⊗ F, Rd ), and Lp (R+ × , F , Rd ), respectively. Therefore, Js,t [S()] and Js,t [SF ()] are nonempty subsets of the space Lp (, F, Rd ). Sets Js,t [S()] and Js,t [SF ()] are said to be Aumann stochastic functional integrals of the set-valued stochastic process , over the interval [s, t]. It is clear that Js,t [SF ()] ⊂ Js,t [S()], because SF (F ) ⊂ S(). Aumann stochastic functional integrals can be also defined for nonempty subsets of spaces Lp (R+ × , β+ ⊗ F, Rd ) and Lp (R+ × , F , Rd ), respectively, and their properties follow from properties of Aumann integrals presented above. In particular, it can be verified that for every nonempty decomposable set K ⊂ Lp (R+ × , β+ ⊗ F, Rd ), the Aumann stochastic functional integral Js,t (K) is a decomposable subset of the space Lp (, F, Rd ). Indeed, let u, v ∈ Js,t (K). By the definition of Js,t (K) there are f, g ∈ K such that u = Js,t (f ) and v = Js,t (g). By the decomposability of K ⊂ Lp (R+ × , β+ ⊗ F, Rd ) for every C ∈ β+ ⊗ F we have 1C f + 1C ∼ g ∈ K, where C ∼ = (R+ × ) \ C. In particular, for every A ∈ F we have [s, t] × A ∈ β+ ⊗ F. Therefore, 1[s,t]×A f + 1([s,t]×A)∼ g ∈ K, and then Js,t (1[s,t]×A f + 1([s,t]×A)∼ g) ∈ Js,t (K). But Js,t (1[s,t]×A f + 1([s,t]×A)∼ g) = Js,t (1[s,t]×A f ) + Js,t (1([s,t]×A)∼ g) and ([s, t] × A)∼ = [0, s) × A∼ ∪ (t, ∞) × A∼ . Therefore, Js,t (1[s,t]×A f +1([s,t]×A)∼ g) = Js,t (1A f )+Js,t (1A∼ g) = 1A Js,t (f )+ 1A∼ Js,t (g) = 1A u + 1A∼ v. Then for every A ∈ F and u, v ∈ Js,t (K) one has 1A u + 1A∼ v ∈ Js,t (K). Unfortunately, the last property of Aumann stochastic functional integrals does not hold true for nonempty decomposable subsets of the space Lp (R+ × , F , Rd ). Indeed, let (t, ω) = : [0, 1] for t ∈ [0, 1], and ω ∈ , and let K = SF (), and J (K) = J0,1 (K). Suppose J [SF ()] is decomposable. Then for every A ∈ F1 and ϕ1 , ϕ2 ∈ SF (), one has 1A J (ϕ1 ) + 1A∼ J (ϕ2 ) ∈ J [SF ()], where A∼ =  \ A. Taking in particular, ϕ1 = 1, ϕ2 = 0 we get 1A J (ϕ1 ) ∈ J [SF ()]. Let ψ = 1A×[0,1] for A ∈ F1 \ Ft with fixed t ∈ [0, 1). We have J (ψ) = 1A J (ϕ1 ), which implies that J (ψ) ∈ J [SF ()]. Then we

4.2 Aumann Stochastic Integrals

119

get ψ ∈ SF (), and J (ψ) ∈ J [SF ()]. Contradiction. Thus, J [SF ()] is not decomposable. The following properties of Aumann stochastic functional integrals follow immediately from properties of Aumann integral. Theorem 4.2.1 For every nonempty sets K, H ⊂ Lp (R+ × , β+ ⊗ F, Rd ) or K, H ⊂ Lp (R+ × , F , Rd ) one has: (i) cl[Js,t (K)] = cl[Js,t (K)] and cl[Js,t (co K)] = co Js,t (K), where closures are taken with respect to the norm topologies of Lp (, F, Rd ) and Lp (R+ × , F , Rd ), respectively, (ii) cl[Js,t (K + H)] = cl[Js,t (K) + Js,t (H)], (iii) if K ⊂ Lp (R+ ×, β+ ⊗F, Rd ) is bounded decomposable, then Js,t (clw (H)) is a convex weakly compact subset of L2 (, Ft , Rd ), (iv) if K ⊂ Lp (R+ × , β+ ⊗ F, Rd ) is closed decomposable and integrably bounded, then lim|t−s|→0 H (J0,t (K), J0,s (K)) = 0, for every 0 ≤ s < t < ∞, (v) if K ⊂ Lp (R+ × , β+ ⊗ F, Rd ) is decomposable, then Js,t (K) is integrably bounded if and only if K is bounded.   Differently to Aumann stochastic functional integrals, Aumann stochastic integrals are defined as set-valued mappings with values inthe space Cl(Rd ). We begin t with the definition of Aumann stochastic integral (A) s K dτ defined, for a given nonempty set K ⊂ Lp (R+ × , β+ ⊗ F, Rd ) and fixed 0 ≤ s < t < ∞, t t by setting (A) s K dτ :   ω → cl{ s ϕ(τ, ω)dτ : ϕ ∈ K} ∈ Cl(Rd ) for every  t ω ∈ . Hence in particular, it follows that if K is a countable set, then (A) s K dτ is a set-valued random variable. Indeed, in such case there is a sequence ⊂ Lp (R+ × , β+ ⊗ F, Rd ) such that K = {ϕn : n ≥ 1}. Therefore, (ϕn )∞ n=1 t t ((A) s K dτ )(ω) = cl{ s ϕn (τ, ω)dτ : n ≥ 1}. Hence, by Theorem 2.2.3 of Chapt ter 2, it follows that (A) s K dτ is F-measurable. In particular, if K = S(F ), where S(F ) is a set of all measurable selectors of a measurable p-integrably bounded t multifunction of F : R+ × → Cl(Rd ), a set-valued stochastic integral (A) s K dτ t is denoted by (A) s Fτ dτ and said to be the Aumann stochastic integral of F over the interval [s, t]. If a multifunction F : R+ ×  → Cl(Rd ) isF-non-anticipative t p-integrably bounded, then the Aumann stochastic integral (A) s Fτ dτ is denoted t t t by (AF ) s Fτ dτ and defined by setting ((AF ) s Fτ dτ )(ω) = cl{ s ϕ(τ, ω)dτ : ϕ ∈ SF (F )} for every ω ∈ . The second type of Aumann stochastic integrals of multifunction F can be defined immediately by the Aumann integral depending on a random parameter. Moreprecisely, for a given above multifunction  t F the sett valued mapping   ω → s F (τ, ω)dτ ∈ Cl(Rd ) is denoted by s F (τ, ·)dτ t or by s Fτ dτ and called the Aumann stochastic integral depending on a random t parameter. From this definition it follows that s F (τ, ω)dτ = Js,t (S(·, ω)) for every ω ∈ , where S(·, ω) denotes subtrajectory integrals of the set-valued d mapping F (·, ω) : R+  t → F (t, ω) ∈ Cl(R  t ) for every ω  t ∈ . But, for every v ∈ S(F ) one has v(·, ω) ∈ S(·, ω). Then s v(τ, ω)dτ ∈ s F (τ, ω)dτ for t t every ω ∈ , which implies that cl{ s v(τ, ω)dτ : ϕ ∈ S()} ⊂ s F (τ, ω)dτ

120

4 Aumann Stochastic Integrals

t t for every ω ∈ . Thus, ((A) s Fτ dτ )(ω) ⊂ s F (τ, ω)dτ for every ω ∈ . t t Similarly, the inclusion ((AF ) s Fτ dτ )(ω) ⊂ s F (τ, ω)dτ for every ω ∈ , for an F-non-anticipative p-integrably bounded multifunction F : R+ ×  → Cl(Rd ), can be obtained. From  t properties of Aumann integrals it follows that the Aumann stochastic integral s F (τ, ·)dτ is a set-valued random variable. It is a consequence of the following theorem. Theorem 4.2.2 If F : R+ ×  → Cl(Rd ) is a measurable (F-non-anticipative) integrably  t bounded set-valued stochastic process, then the set-valued mapping  ∈ ω → 0 F (τ, ω)dτ ∈ Cl(Rd ) is F-measurable (Ft -measurable) for every t ≥ 0. Proof Let t ≥ 0 be fixed. 4.1.7, we can assume that F is  t By (iv) of Theorem t convex-valued, because 0 F (τ, ω)dτ = 0 co F (τ, ω)dτ for every ω ∈ . By [38, Th. 3.8 of Chap. II], a function ϕ : R+ ×   (t, ω) → s(p, F (t, ω)) ∈ R t is β+ ⊗ F-measurable. Then a function   ω → 0 s(p, F (τ, ω))dτ ∈ R is d F-measurable for every 4.1.7, for every p ∈ Rd  t p ∈ R . By (iii) of Theorem t and ω ∈  one has 0 s(p, F (τ, ω))dτ = s(p, 0 F (τ, ω))dτ ). Then the function t   ω → s(p, 0 F (τ, ω))dτ ) ∈ R is F-measurable, which by [38, Theorem 3.8 of t Chap. II] implies that the set-valued mapping  ∈ ω → 0 F (τ, ω)dτ ∈ Cl(Rd ) is F-measurable. If F : R+ ×  → Cl(Rd ) is F-non-anticipative, then it is measurable and Fadapted, which implies that ϕτ :   ω → s(p, F (τ, ω)) ∈ R is Fτ -measurable for every p ∈ Rd and τ ∈ [0, t]. In particular, s(p,  t Fτ (·)) is Ft -measurable for every p ∈ Rd . Then the function   ω → 0 s(p, F (τ, ω))dτ ∈ R is d Ftt -measurable for every p ∈t R . Hence, similarly as above due to the equality 0 s(p, F (τ,ω))dτ = s(p, 0 F (τ, ω))dτ ), it follows that a set-valued mapping t    ∈ ω → 0 F (τ, ω)dτ ∈ Cl(Rd ) is Ft -measurable for every t ≥ 0. d Corollary 4.2.1 If F : R+ ×  → Cl(R  t ) is measurable and integrably bounded, then the set-valued stochastic process ( 0 Fτ dτ )t≥0 is continuous.

Proof By Lemma 3.3.4 of Chapter 3, for every t0 ∈ [s, t] one has S(F ) = 1[0,t0 ] S(F ) + 1[t0 ,t] S(F ) for every ω ∈ . Then J0,t (S(F )) = J0,t0 (S(F )) + t t t Jt0 ,t (S(F ), which implies that 0 Fτ dτ = 00 Fτ dτ + t0 Fτ dτ . Thus, for every t  t0 t ω ∈  one has 0 Fτ (ω)dτ = 0 Fτ (ω)dτ + t0 Fτ (ω)dτ . Therefore, for every t0 ∈ (s, t), and every sequence (tn )∞ ⊂ [t0 , t) convergent to t0 , we obtain  tn  t0  t0n=1 t t h( 0 Fτ (ω)dτ, 0 Fτ (ω)dτ ) = h( 0 Fτ (ω)dτ + t0n Fτ (ω)dτ, 00 Fτ (ω)dτ ) ≤  tn for every sequence (tn )∞ n=1 ⊂ (s, t0 ] t0 Fτ (ω)dτ for every ω ∈ . Similarly,  t0 t tn convergent to t0 , one gets h( 0 Fτ (ω)dτ, 0 Fτ (ω)dτ ) ≤ tn0 Fτ (ω)dτ . Then





tn 0

for n ≥ 1. Thus, h(

Fτ (ω)dτ 0

 tn 0

Fτ (ω)dτ,



t0

Fτ (ω)dτ,

h

 t0 0

  ≤ 

tn t0

  Fτ (ω)dτ 

Fτ (ω)dτ ) → 0 as n → ∞ for every ω ∈ .

 

4.3 Lebesgue Set-Valued Stochastic Integrals

121

Remark 4.2.1 Similarly as above, it followsthat if F : R+ × → Cl(Rd ) is F-nont anticipative and integrably bounded, then ( 0 Fτ dτ )t≥0 is a continuous F-adapted set-valued stochastic process, because similarly to the proof of Lemma 3.3.4 of Chapter 3, we can get SF (F ) = 1[0,t0 ]× SF (F )) + 1[t0 ,t]× SF (F )), which implies t t t that 0 F (τ, ω)dτ = 00 F (τ, ω)dτ + t0 F (τ, ω)dτ for every ω ∈ .  

4.3 Lebesgue Set-Valued Stochastic Integrals Aumann stochastic functional integrals can be applied to the definition of the Lebesgue set-valued stochastic integral. More precisely, for a given nonempty set K ⊂ Lp (R+ × , β+ ⊗ F, Rd ) or K ⊂ Lp (R+ × , F , Rd ), the Lebesgue  t setvalued stochastic integral of the set K over the interval [s, t], denoted by () s K dτ , t is defined as a set-valued random variable such that the set SF (() s K dτ ) of t all F-measurable selectors of () s K dτ is equal to the set decF (Js,t (K)). If d stochastic integral is K ⊂ Lp (R+ ×  t, F , R ), then its Lebesgue set-valued t denoted by (F ) s K dτ , and defined by a set SFt [(F ) s K dτ ] of all Ft -measurable t selectors of (F ) s Kdτ . It has to be equal to decFt (Js,t (K)). Similarly as above, for given a measurable or an F-non-anticipative integrably bounded set-valued  t process F : R+ ×  → Cl(Rd ), Lebesgue set-valued stochastic integral () s S(F )dτ , t t t and (F ) s SF (F )dτ are denoted by () s Fτ dτ and (F ) s Fτ dτ , respectively, and said to be the Lebesgue set-valued stochastic integral of F over the interval [s, t]. From the above definition and the definition of the  t Aumann tstochastic integral depending on random parameter, it follows that () s Fτ dτ ⊂ s Fτ dτ a.s. Indeed, t for every u ∈ Js,t (S(F )) there is f ∈ S(F ) such that u(ω) = s fτ (ω)dτ for t t every ω ∈ . Then u(ω) ∈ s F (τ, ω)dτ = ( s Fτ dτ )(ω) for every ω ∈ , t t which implies that u ∈ SF ( s Fτ du). Thus, Js,t (S(F )) ⊂ SF ( s Fτ du). Therefore, t t decF (Js,t (S(F ))) ⊂ SF ( s Fτ du), because SF ( s Fτ du) is a closed decomposable t subset of the space Lp (, F, Rd ). Hence, it follows that SF (() s Fτ dτ ) ⊂ t t SF ( s Fτ du), which by Corollary 2.3.1 of Chapter 2 implies that () s Fτ dτ ⊂ t bounded set-valued s Fτ dτ a.s. Similarly, for an F-non-anticipative p-integrably t t process F : R+ ×  → Cl(Rd ) we obtain (F ) s Fτ dτ ⊂ s Fτ dτ a.s. For a measurablep-integrably bounded set-valued process F : R+ ×  → Cl(Rd ) we t t also get () s Fτ dτ ⊂ (A) s Fτ dτ a.s. Indeed, it is clear that S(F ) = ∅, and that t for every u ∈ Js,t (S(F )), there is f ∈ S(F ) such that u(ω) = s f (τ, ω)dτ for t t every ω ∈ . But, for every ω ∈  one has ((A) s Fτ dτ )(ω) = cl{ s f (τ, ω)dτ : t f ∈ S(F )}. Therefore, for every u ∈ Js,t (S(F )) one has u ∈ SF ((A) s Fτ dτ ). t Then Js,t (S(F )) ⊂ SF ((A) s Fτ dτ ), which implies that decFt [Js,t (S(F )) ⊂ t SF ((A) s Fτ dτ ). Hence, by the definition of the Lebesgue set-valued stochastic t t integral, it follows that SF (() s Fτ dτ ) ⊂ SF ((A) s Fτ dτ ), which by Corolt t lary 2.3.1 of Chapter 2 implies that () s Fτ dτ ⊂ (A) s Fτ dτ a.s. In a similar way,

122

4 Aumann Stochastic Integrals

we can verify that for every F-non-anticipative p-integrably bounded t  t set-valued mapping F : R+ ×  → Cl(Rd ), we also have (F ) s Fτ dτ ⊂ (AF ) s Fτ dτ a.s. The following theorem follows immediately from above definitions. t Theorem 4.3.1 If K ⊂ Lp (R+ × , β+ ⊗ F, Rd ), then (() s K dτ ))(ω) = t (() s co K dτ )(ω) for a.e. ω ∈ . If K ⊂ Lp (R+ × , F , Rd ) is decomposable, t t then ((F ) s co Kdτ )(ω) = co {(F ) s K dτ ))(ω)} for a.e. ω ∈ . Proof Let K ⊂ Lp (R+ × , β+ ⊗ F, Rd ) be given. By Theorems 4.1.2 and 4.1.1, one has cl[Js,t (K)] = cl[Js,t (co K)] = cl[Js,t (co K)]. It is clear that SF (cl(Js,t (K)) = SF (Js,t (K)), and SF (cl(Js,t (co K))) = SF (Js,t (co K)). t t Then SF (() s K dτ ) = SF (() s co K dτ ), which by Corollary 2.3.1 of t t Chapter 2 implies that () s K dτ = () s co K dτ a.s. By Theorem 4.2.1, and Theorem 3.3.3 of Chapter 3, one has cl{Js,t (K)} = cl{Js,t (co K)} = co{Js,t (K)} and decF {Js,t (co K)} = co{decF (Js,t (K))}. Hence, by the definition of Lebesgue set-valued 2, it follows  t stochastic integrals, and t Theorem 2.3.5 of Chapter t SF ((F ) s co Kdτ ) = co{SF ((F ) s K dτ )} = SF (co (F ) s K dτ ), which by t t Corollary 2.3.1 of Chapter 2 implies that (F ) s co K dτ = co{(F ) s K dτ } a.s.  

4.4 Approximation of Aumann Stochastic Integrals We shall now present some approximation theorems for Aumann and Lebesgue setvalued stochastic integrals. They will be presented first for continuous bounded, and continuous integrably bounded set-valued stochastic processes. Later on, for measurable and F-non-anticipative integrably bounded set-valued stochastic processes, the problem will be considered. We begin with the following lemma. Lemma 4.4.1 If  : R+ ×  → Cl(Rd ) is a continuous bounded set-valued stochastic process, then for every fixed 0 ≤ s < t < ∞, and every sequence r r r (πr )∞ r=1 of partitions πr = {s = t0 < t1 < . . . < tlr = t} of the r r interval [s, t] such that max0≤i≤lr −1 (ti+1 − ti ) → 0 as r → ∞, one has  t lr −1 t r ) (τ )t r dτ, 1[tir ,ti+1 limr→∞ Eh( s i=0 s τ dτ ) = 0. i Proof Let 0 ≤ s < t < ∞, and a sequence (πr )∞ r=1 be given. By Lemma 3.3.4 of Chapter 3, for every r ≥ 1 we get

 Eh s

 Eh s

r −1 t l

i=0

r −1 t l

 r ) (τ )t r dτ, 1[tir ,ti+1 i

i=0

 r ) (τ )t r dτ, 1[tir ,ti+1 i

r −1 t l

s

i=0



t

τ dτ

=

s

r ) (τ )τ dτ 1[tir ,ti+1

≤

4.4 Approximation of Aumann Stochastic Integrals



r −1 t l

s

r ) (τ ) 1[tir ,ti+1

i=0

123

max

u,v∈[s,t],|u−v|≤δr

E[h(u , v )]dτ,

r where δr = max0≤i≤lr −1 (ti+1 − tir ). But the set-valued mapping [s, t]  τ → τ (ω) ⊂ Rd is bounded and uniformly continuous for every fixed ω ∈ . Then maxu,v∈[s,t],|u−v|≤δr E[h(u , v )] → 0 as r → ∞. Therefore, for every ε > 0 there is rε ≥ 1 such that maxu,v∈[s,t],|u−v|≤δr E[h(u , v )] ≤ ε/t for every r ≥ rε . Then





r −1 t l

Eh

1

r ) [tir ,ti+1

s

i=0



t

(τ ) dτ, tir

 ≤ ε/t

τ dτ s

r −1 t l

s

r ) (τ )dτ ≤ ε 1[tir ,ti+1

i=0

 

for every ε > 0 and r ≥ rε . We can now prove the following approximation theorem.

Theorem 4.4.1 If the assumptions of Lemma 4.4.1 are satisfied, then for every 0 ≤ r r s < t < ∞ and every sequence (πr )∞ r=1 of partitions πr = {s = t0 < t1 < . . . < r r r tlr = t} of the interval [s, t] such that max0≤i≤lr −1 (ti+1 − ti ) → 0 as r → ∞, one t lr −1 r r has limr→∞ Eh( s τ dτ, i=0

ti · co(tir )) = 0, where tir = ti+1 − tir for every i = 0, 1, . . . , lr and r ≥ 1. Proof The result follows immediately from Lemma 4.4.1, and the equality



r −1 t l

Eh s

r ) (τ )t r dτ, 1[tir ,ti+1 i

i=0

l r −1



tir

· co(tir ) = 0,

(4.4.1)

i=0

satisfied for every r ≥ 1. To see that (4.4.1) is satisfied, let us note that by (v) of The t −1  t lr −1 r ) (τ )t r dτ = r r r 1[tir ,ti+1 orem 4.1.7, one has s i=0 i=0 1[ti ,ti+1 ) (τ )co(ti )dτ . s i Therefore, it is enough only to verify that (4.4.1) is satisfied by the assumption that  is convex-valued. Let us note that by properties of the Hausdorff metric (see Lemma 1.3.3 of Chapter 1), Lemmas 4.4.1 and 4.1.5, one gets



r −1 t l

Eh s l r −1

Eh

i=0



r ti+1

tir

r ) (τ )t r dτ, 1[tir ,ti+1 i

i=0



tir

· tir

≤

i=0

 tir dτ, tir · tir

l r −1

=

l r −1

 Eh tir · tir , tir · tir =0

i=0

for every r ≥ 1. Thus, (4.4.1) is satisfied for every r ≥ 1. R+

 

Lemma 4.4.2 If  : × → is a continuous bounded set-valued t stochastic process, then for every fixed 0 ≤ s < t < ∞ one has () s τ dτ = t s τ dτ a.s. Cl(Rd )

124

4 Aumann Stochastic Integrals

Proof By the general relationships  t between tAumann and Lebesgue set-valued stochastic integrals we have () s τ dτ ⊂ s τ dτ a.s. To obtain the converse inclusion, let us note that by Theorem 2.3.4, and Remark 2.3.2 of Chapter 2, we get

 t  

 t 

 t τ dτ , SF () τ dτ =H¯ SF τ dτ , decF Js,t (S()) H¯ SF s

s



≤ H¯ SF



s



t s

τ dτ , SF

l r −1

(4.4.2)



tir · co(tir ) +

i=0



 t

l l r −1 r −1

tir · co(tir ) , decF Js,t (S(tir )) ≤ E h¯ τ dτ,

tir · co(tir ) H¯ SF s

i=0

i=0



l r −1 ¯ r ) SF (co t r ) , Js,t (S()) , 1[tir ,ti+1 +H Js,t i i=0

where (πr )r≥1 is the sequence of partitions πr = {s = t0r < t1r < . . . < r tlrr = t} of the interval [s, t] such that max0≤i≤lr −1 (ti+1 − tir ) → 0 as t lr −1 r r → ∞. By Theorem 4.4.1, one has limr→∞ Eh( s τ dτ, i=0

ti · co(tir )) = 0, a.s. For simplicity, similarly as above, we can assume that  is convex-valued. To conclude the proof it is enough only to verify that  ¯ s,t [ lr −1 1[t r ,t r ) SF (t r )], Js,t (S())) = 0. To see that, let us limr→∞ h(J i=0 i i+1 i define for every r ≥ 1 the multifunction r : [s, t] ×  → Cl(Rd ) by setting lr −1 r r r r (τ, ω) = i=0 1[ti ,ti+1 ) (τ )ti (ω) for every (τ, ω) ∈ [s, t] × . It is clear that lr −1 r r r i=0 1[ti ,ti+1 ) SF (ti ) ⊂ S( r ) for every r ≥ 1. Now, by Theorem 2.3.4 and Remark 2.3.2 of Chapter 2, one gets

l  r −1 ¯h Js,t ¯ s,t [S( r r r 1[ti ,ti+1 ) SF (ti ) , Js,t (S()) ≤ h(J

r )], Js,t [S()])

i=0

    t  t    sup inf E  uτ dτ − vτ dτ  : v ∈ S() : u ∈ S( 

s

 ≤

r)

s

   t  ≤ sup inf E |uτ − vτ |dτ : v ∈ S() : u ∈ S( s

 E s

t



l r −1 ¯h r r r 1[ti ,ti+1 ) (τ )ti , τ dτ. i=0

 r)

=

=

4.4 Approximation of Aumann Stochastic Integrals

125

Similarly to the proof of Lemma 4.4.1, it follows that for every ε > 0 there is rε ≥ 1 such that for every r ≥ rε , one has E h¯



r −1 t l

s

 r ) t r dτ, 1[tir ,ti+1 i



t

τ dτ



s

i=0

r −1 t l

≤ ε/t s

r ) (τ )dτ = ε. 1[tir ,ti+1

i=0

lr −1 r ) SF (t r )], Js,t (S())) = 0. Hence, it 1[tir ,ti+1 Therefore, limr→∞ H¯ (Js,t [ i=0 i t t t follows that H¯ (SF [ s τ dτ ], SF [() s τ dτ ]) = 0. Therefore, SF [ s τ dτ ] ⊂ t t SF [() s τ dτ ], which by Corollary 2.3.1 of Chapter 2, implies that s τ dτ ⊂ t   () s τ dτ a.s. Lemma 4.4.3 If  : R+ ×  → Cl(Rd ) is a continuous bounded and an Fadapted  t set-valued  t stochastic process, then for every fixed 0 ≤ s < t < ∞ one has (F ) s τ dτ = s τ dτ a.s. t t Proof Similarly as above we have (F ) s τ dτ ⊂ s τ dτ a.s. To obtain the converse inclusion, for simplicity let us assume that  is convex-valued, and let us note that



 t

 t  t  t ¯ ¯ τ dτ , SFt (F ) τ dτ = Eh τ dτ, (F ) τ dτ ≤ H SFt s

E h¯



s

t

τ dτ, s

l r −1



tir · tir

s

+ E h¯

i=0

l r −1

s



tir · tir , (F )

t s

(τ, ·)dτ + r

s

i=0

 ¯ E h (F )

t

 t τ dτ , r (τ, ·)dτ, (F ) s

where ({s = t0r < t1r < . . . < tlrr = t})r≥1 is the sequence of partitions of r the interval [s, t] such that max0≤i≤lr −1 (ti+1 − tir ) → 0 as r → ∞, and r : lr −1 r ) (τ )t r (ω) 1[tir ,ti+1 [s, t] ×  → Cl(Rd ) is defined by setting r (τ, ω) = i=0 i for every r ≥ 1 and (τ, ω) ∈ [s, t] × . By Theorem 4.4.1, we have t lr −1 r ¯ r limr→∞ E h( = 0. Furthermore, by virtue of i=0 ti · ti ) s τ dτ, Lemma 3.3.4 of Chapter 3, one gets

 E h¯ (F ) l r −1 i=0

 Eh (F )

r ti+1

tir

t s

 t (τ, ·)dτ, ( )  dτ ≤ r τ F

 tir dτ, (F )

s

r ti+1

tir

τ dτ

≤

l r −1  t r i+1 i=0

tir

Eh(tir , τ )dτ.

126

4 Aumann Stochastic Integrals

r r lr −1  ti+1 lr −1  ti+1 But, i=0 Eh(tir , τ )dτ ≤ i=0 maxu,v∈[s,t],|u−v|≤δr E[h(u , v )], tir tir r r τ )dτ , where δr = max0≤i≤lr −1 (ti+1 − ti ). Similarly to the proof of Lemma 4.4.1,   ¯ F ) t r (τ, ·)dτ, (F ) t τ dτ ) = 0. hence it follows that limr→∞ E h(( s s  ¯ lr −1 t r · Therefore, to conclude the proof, one has to verify that E h( i i=0 t tir , (F ) s r (τ, ·)dτ ) = 0. Let us note that

E h¯

l r −1



tir · tir , (F )

i=0

t

r (τ, ·)dτ

≤

s

l r −1

 E h¯ tir · tir , (F )

r ti+1

tir

i=0

tir dτ ,

r ], β r , dt) and to and by Lemma 4.1.5, applied to a measurable space ([tir , ti+1 i 1 d the closed convex set SFt r (tir ) ⊂ L (, Ftir , R ) with the Borel σ -algebra i

r ], for every i = 1, 2, . . . , l − 1 and r ≥ 1, we βir of the interval [tir , ti+1 r r  ti+1 r get t r SFt r (tir )dτ = ti · SFt r (tir ). By the definition of the Aumann i i i  tr  tr stochastic integral t ri+1 SFt r (tir )dτ it follows that t ri+1 SFt r (tir )dτ = i i i i r (Sβ r [SF r (t r )]) for every i Jtir ,ti+1 = 1, 2, . . . , lr − 1 and r ≥ 1. For ti i i r (Sβ r [SF r (t r )]) there is v every u ∈ Jtir ,ti+1 ∈ Sβir [SFt r (tir )] such that ti i i i r (v). By Theorem 3.1.6 of Chapter 3, for every v ∈ Sβ r [SF r (t r )] u = Jtir ,ti+1 ti i i there is the uniquely determined (except for a set of dt × P -measure zero) r ] ×  → Rd such that (βir ⊗ Ftir , β(Rd ))-measurable function f : [tir , ti+1 r r r r ] be the set of the f (t, ·) = v(t) for a.e. t ∈ [ti , ti+1 ]. Let Ai ⊂ [tir , ti+1 r ] \ Ar . Let measure zero such that f (t, ·) ∈ SFt r (tir ) for every t ∈ [tir , ti+1 i i r ] \ Ar , and f (t, ·) = ur for t ∈ Ar , with f (t, ·) = f (t, ·) for t ∈ [tir , ti+1 i i i ur ∈ SF r (t r ). We have f (t, ·) ∈ SF r (t r ) for every t ∈ [t r , t r ]. It is clear that i

ti

ti

i

i

i

i+1

r ]× → Rd is (β r ⊗F r , β(Rd ))-measurable and an Fr -adapted, where f : [tir , ti+1 ti i i r r . Then for every u ∈ Jt r ,t r (Sβ r [SF r (t r )]) there exists an Fr Fi = (Ft )tir ≤t≤ti+1 i ti i i+1 i i r ] ×  → Rd such that u = J r r (f ), and non-anticipative function f : [tir , ti+1 ti ,ti+1 f (t, ·) ∈ SF r (t r ) for every t ∈ [t r , t r ]. Therefore, for every set A ∈ β r ⊗ Ft r ti

i

i

i+1

i

i

and t ∈ one has 1A (t, ·)f (t, ·) ∈ 1A (t, ·)SFt r (tir ), which implies i r r  ti+1  ti+1 that E t r 1A (t, ·)f (t, ·)dt ∈ E t r 1A (t, ·)SFt r (tir )dt. But SFt r (tir ) ⊂ i i i i   dtdP ∈ r ⊗F r (t r ), which by the definiSβir ⊗Ft r (tir ). Therefore, S f β A A i i   i ti  (t, ω)dtdP ∈ r tion of the Aumann integral it follows that f A A ti (ω)dtdP  r r for every A ∈ βi ⊗ Fti . Hence it follows that s(p, A f (t, ω)dtdP ) ≤  s(p, A tir (ω)dtdP ) for every p ∈ Rd and A ∈ βir ⊗ Ftir , which implies that   s(p, A f (t, ω)dtdP ) − s(p, A tir (ω)dtdP ) < ε for every ε > 0 p ∈ Rd and A ∈ βir ⊗ Ftir . Therefore, for every ε > 0 p ∈ Rd and A ∈ βir ⊗ Ftir   we obtain max|p|=1 {s(p, A f (t, ω)dtdP ) − s(p, A tir (ω)dtdP )} ≤ ε, r ], [tir , ti+1

4.4 Approximation of Aumann Stochastic Integrals

127

  ¯ r which is equivalent to h( A f (t, ω)dtdP , A ti (ω)dtdP ) ≤ ε for every ε > 0 and A ∈ βir ⊗ Ftir . Then for every A ∈ βir ⊗ Ftir we have   r r A f (t, ω)dtdP ∈ A ti (ω)dtdP , which implies that f (t, ω) ∈ ti (ω) r r r for a.e. (t, ω) ∈ [ti , ti+1 ] × . But, f is Fi -non-anticipative process. Therefore, f ∈ SFri (tir ) for every i = 1, 2, . . . , lr − 1 and r ≥ 1. Thus, for every ∈ SFr (t r ) such that u = Jt r ,t r (f ) ∈ r (Sβ r [SF r (t r )]) there is f u ∈ Jtir ,ti+1 ti i i i i i i+1 r (SF r (t r )). Therefore, Jt r ,t r (Sβ r [SF r (t r )]) r ,t r (SF r (t r )) Jtir ,ti+1 ⊂ J t ti ti ti i i i+1 i i i i+1 i for every i = 1, 2, . . . , lr − 1 and r ≥ 1. But, by Lemma 4.1.5, for every r (Sβ r [SF r (t r )]) = SF r ( t r · t r ). i = 1, 2, . . . , lr − 1, and r ≥ 1 we have Jtir ,ti+1 i t i i i i i

r (SF r (t r )) ⊂ decF r {Jt r ,t r (SF r (t r ))} = Therefore, SFir ( tir · tir ) ⊂ Jtir ,ti+1 ti ti+1 ti i i i+1 i r  ti+1 SFt r ((F ) t r tir dτ ), which by Corollary 2.3.1 of Chapter 2, implies that i i  tr r and r ≥ 1. Then

tir · tir ⊂ (F ) t ri+1 tir dτ for every i = 0, 1, . . . , li−1 i  lr −1 r t ¯ r   E h( r (τ, ·)dτ ) = 0 for every r ≥ 1. i=0 ti · ti , (F ) s

Corollary 4.4.1 If  : R+ ×  → Cl(Rd ) is a continuous bounded set-valued t stochastic process, then for every fixed 0 ≤ s < t < ∞ one has () s τ dτ = t t t (A) s τ dτ = s τ dτ a.s. If furthermore,  is F-adapted, then (F ) s τ dτ = t t (AF ) s τ dτ = s τ dτ a.s. Proof By relationships between Aumann and Lebesgue t  t set-valued tstochastic integrals presented above, we have () s τ dτ ⊂ (A) s τ dτ ⊂ s τ dτ a.s. By t t Lemma 4.4.2 (Lemma 4.4.3) we have () s τ dτ = s τ dτ a.s., which implies t t t that () s τ dτ = (A) s τ dτ = s τ dτ a.s. If  is F-adapted, then in a similar t t t   way we obtain (F ) s τ dτ = (AF ) s τ dτ = s τ dτ a.s. We extend now the above results to the case of measurable (F-non-anticipative) bounded convex-valued stochastic processes. Theorem 4.4.2 If  : R+ ×  → Cl(Rd ) is measurable (F-non-anticipative) bounded convex-valued set-valued process, then for every fixed 0 ≤ s < t < ∞ such that E(t, ·) < ∞ there exists the set H ⊂ [s, t] of the Lebesgue measure zero such that for every sequence (πr )r≥1 of partitions πr = {s = t0r < t1r < . . . < tlrr = t} of the interval [s, t], with {t0r , t1r , . . . , tlrr } ⊂ [s, t] \ H for every r ≥ 1, t lr −1 r such that δr → 0 as r → ∞, one has limr→∞ Eh( s τ dτ, i=0

ti · tir ) = 0, r r where δr = max0≤i≤lr −1 (ti+1 − tir ) and tir = ti+1 − tir for i = 0, 1, . . . , lr − 1 and r ≥ 1. Proof Let 0 ≤ s < t < ∞ be fixed and m > 0 be such that τ (ω) ≤ m for a.e. (τ, ω) ∈ [s, t] × . By Theorem 4.1.10, there exists a sequence (n )∞ n=1 of continuous bounded set-valued processes n : [0, t] ×  × Cl(Rd ) with compact convex values such that nτ (ω) ≤ m and h(nτ (ω), τ (ω)) → 0 for fixed ω ∈ , t and a.e. τ ∈ [0, t] as n → ∞. Then limn→∞ s Eh(nτ , τ )dτ = 0. Taking an

128

4 Aumann Stochastic Integrals

appropriate subsequence if needed, we can assume that Eh(nτ , , τ ) → 0 for a.e. τ ∈ [s, t] as n → ∞. Let H ⊂ [s, t] be a set of the Lebesgue measure zero such that limn→∞ Eh(nτ , τ ) = 0 for every τ ∈ [s, t] \ H and let (πr )r≥1 be a sequence of partitions πr = {s = t0r < t1r < . . . < tlrr = t} of the interval [s, t], with r −tr) → {t0r , t1r , . . . , tlrr } ⊂ [s, t] \ H for every r ≥ 1 and such that max0≤i≤lr −1 (ti+1 i 0 as r → ∞. For every n ≥ 1 one gets



t

Eh

τ dτ, s

l r −1





t

Eh s

nτ dτ,

l r −1

s



tir

+ Eh

· ntr i

i=0

t

τ dτ,

i=0





t

≤ Eh

tir · tir

s

l r −1

tir

nτ dτ +

· ntr , i

i=0

l r −1

tir

(4.4.3) · . tir

i=0

But, by (vi) of Theorem 4.1.7, and Lemma 1.3.3 of Chapter 1, for every n ≥ 1 and r ≥ 1, one has



t

Eh s

 nτ dτ,



t

τ dτ



t

≤E

s

h(n (τ, ω), (τ, ω))dτ,

s

and Eh

l r −1

tir

i=0

· ntr , i

l r −1



tir

·

≤

tir

i=0

l r −1

tir Eh(ntr , tir ) i

i=0

t t lr −1 r

ti · for every n, r ≥ 1. Then Eh( s τ dτ, s nτ dτ ) → 0 and Eh( i=0 lr −1 r lr −1 r ntr , i=0

ti · tir ) → 0 as n → ∞, because limn→∞ i=0

ti Eh(n (tir , ·), i (tir , ·)) = 0 for fixed r ≥ 1. Thus, immediately from (4.4.3) it follows that



t

Eh

τ dτ, s

l r −1



tir

· tir



t

≤ limn→∞ Eh s

i=0

nτ dτ,

l r −1 i=0



tir

· ntr i

for every r ≥ 1. But, for every n ≥ 1 we have



t

Eh s l r −1 i=0

tir

nτ dτ,

l r −1 i=0

sup u,v∈[s,t],|u−v|≤δr



tir

· ntr i

≤

l r −1  t r i+1 i=0

E[h(nu , nv )] = t

tir

Eh(nτ , ntr )dτ ≤ i

sup u,v∈[s,t],|u−v|≤δr

E[h(nu , nv )].

4.4 Approximation of Aumann Stochastic Integrals

129

Furthermore, for every n ≥ 1 there exists a positive integer rn ≥ 1 such that δr ≤ 1/n2 for every r ≥ rn . Therefore,

 Eh s

t

nτ dτ,

l r −1



tir · ntr i

i=0

≤t

sup u,v∈[s,t],|u−v|≤1/n2

E[h(nu , nv )]

(4.4.4)

for every n ≥ 1 and r ≥ rn . It is clear that we can select positive integers rn ≥ 1 in such a way that a sequence (rn )∞ n=1 is increasing. Let us note that for 0 ≤ u < v ≤ t such that |u−v| ≤ 1/n2 , one has [u−1/n, u] = [u−1/n, v−1/n]∪[v−1/n, u] and p [v−1/n, v] = [v−1/n, u]∪[u, v], because 1/n2 < 1/n. Let ϕτ (ω) = s(p, τ (ω)) d for every (τ, ω) ∈ [s, t] ×  and p ∈ R such that |p| = 1. By Theorem 4.1.10, we t have nt (ω) = n (t−1/n)∨0 τ (ω)dτ . Therefore, for every 0 ≤ u < v ≤ t such that |u − v| ≤ 1/n2 one has

= nE

E[h(nu , nv )]

nE  nE

(v−1/n)∨0



  sup 

|p|=1

(v−1/n)∨0

sup

|p|=1 (u−1/n)∨0

(u−1/n)∨0

v

n

 

v

−

(v−1/n)∨0 v

|p|=1 u





ϕτp dτ



 |ϕτp |dτ | + nE sup

sup |ϕτp |dτ + nE

(u−1/n)∨0 |p|=1



u

v

 sup |ϕτp |dτ = n

u |p|=1

ϕτp dτ 

≤

 |ϕτp |dτ ≤ (v−1/n)∨0

Eτ dτ +

(u−1/n)∨0

Eτ dτ ≤ 2n m|u − v| ≤ 2m/n,

u p

because sup|p|=1 |ϕτ (ω)| = sup|p|=1 |σ (p, τ (ω))| ≤ τ (ω) ≤ m for a.e. (τ, ω) ∈ [s, t] × . Then supu,v∈[s,t],|u−v|≤1/n2 E[h(nu , nv )] ≤ 2m/n. Similarly, the case of 0 ≤ v < u ≤ t such that |u − v| ≤ 1/n2 can be considered. Hence, t lrn −1 rn due to (4.4.4), it follows that limn→∞ Eh( s nτ dτ, i=0

ti · ntrn ) = 0, which i t lrn −1 rn implies that limn→∞ Eh( s τ dτ, i=0

ti · t rn ) = 0. i Let us note that the above procedure can be applied, starting with an arbitrary taken increasing sequence (rj )∞ j =1 of positive integers. Then for we can select its subsequence (rn )∞ every such sequence (rj )∞ j =1 n=1 such that t lrn −1 rn limn→∞ Eh( s τ dτ, i=0 ti · t rn ) = 0. Thus, for every sequence (πr )r≥1 i of partitions πr = {s = t0r < t1r < . . . < tlrr = t} of the interval [s, t], with r − tr) → 0 {t0r , t1r , . . . , tlrr } ⊂ [s, t] \ H for every r ≥ 1, such that max0≤i≤lr −1 (ti+1 i t lr −1 r as r → ∞, one has limr→∞ Eh( s τ dτ, i=0 ti · tir ) = 0.  

130

4 Aumann Stochastic Integrals

We obtain now the following theorem. Theorem 4.4.3 If  : R+ ×  → Cl(Rd ) is a measurable (F-non-anticipative) bounded convex-valued  t set-valued  t stochastic process, then for every fixed 0 ≤ s < t < ∞ one has () s τ dτ = s τ dτ a.s. Proof Let 0 ≤ s < t < ∞ be fixed. By Theorem 4.1.10, there exists a sequence (n )∞ n=1 of continuous (continuous and F-adapted) bounded convex-valued setvalued stochastic processes n : [s, t] ×  → Cl(Rd ) such that nτ (ω) ≤ m for a.e. (τ, ω) ∈ [s, t] ×  and such that h(nτ (ω), τ (ω)) → 0 for fixed ω ∈  and a.e. τ ∈ [s, t], where m > 0 is such that τ (ω) ≤ m for a.e. (τ, ω) ∈ [s, t] ×. By relationship between Aumann and Lebesgue stochastic integrals one t t has () s τ dτ ⊂ s τ dτ a.s. On the other hand, by Corollary 4.4.1, we have t t () s nτ dτ = s nτ dτ a.s. Hence, and due to Theorem 4.1.7 and properties of the Hausdorff metric one gets h¯

 s

t



 t  t  t τ dτ, () τ dτ ≤ h¯ τ dτ, nτ dτ + s

s

s



 t

 t  t  t nτ dτ, ( ) nτ dτ + h¯ () nτ dτ, () τ dτ ≤ h¯ s

s



t

s

¯ τ , nτ )dτ + h(

s



t s

s

¯ nτ , τ )dτ h(

t t τ dτ ⊂ () s τ dτ a.s. Therefore, () s τ dτ =   t Remark 4.4.1 By the assumptions of Theorem 4.4.3, it follows that () s τ dτ = t t t t t (A) s τ dτ = s τ dτ , and (F ) s τ dτ = (AF ) s τ dτ = s τ dτ ) a.s. for every 0 ≤ s < t < ∞, respectively.   a.s. Hence it follows that t s τ dτ a.s.

t s

Corollary 4.4.2 If  : R+ ×  → Cl(Rd ) is a measurable bounded convexvalued one has  t set-valued stochastic process, then for every fixed 0 ≤ s < t 0 and u ∈ SF [ s τ dτ ] there exists ϕ ∈ S() such that t u(ω) = s ϕτ (ω)dτ for a.e. ω ∈ .

4.4 Approximation of Aumann Stochastic Integrals

131

t Proof By Theorem 4.4.2, and Corollary 4.4.2, for every u ∈ SF [ s τ dτ ], one has t t u ∈ SF [() s τ dτ ]. But SF [() s τ dτ ] = decF Js,t (S()) = cl[Js,t (S())]. t Then for every u ∈ SF [ s τ dτ ] one has u ∈ cl[Js,t (S())], which implies n that there exists a sequence (ϕ n )∞ n=1 of S() such that E|Js,t (ϕ ) − u| → 0 as n → ∞. But, S() is weakly compact subset of the space L1 ([s, t]×, β⊗F, Rd ). n ∞ Therefore, there exists a subsequence (ϕ nk )∞ k=1 of (ϕ )n=1 weakly convergent to any ϕ ∈ S(), which by Dunford–Schwarz theorem implies that a sequence, {Js,t (ϕ nk )}∞ to J (ϕ). But E|Js,t (ϕ nk ) − u| → 0 as k → ∞. k=1 converges weakly t Thus, u = Js,t (ϕ), i.e., u(ω) = s ϕτ (ω)dτ for a.e. ω ∈ .   Remark 4.4.2 A similar result to Corollary 4.4.3 was obtained by Ch. Hess ([2], Th. 2.1) for an unbounded multifunction, by the assumption that (, F, P ) is a separable probability space.   We extend now the above approximation theorem to the case of measurable (Fnon-anticipative) integrably bounded convex-valued stochastic processes. Similarly as above, hence the equalities of the Aumann and the Lebesgue set-valued stochastic integrals for measurable (F-non-anticipative) integrably bounded convex-valued stochastic processes will be followed. Theorem 4.4.4 Let  : R+ ×  → Cl(Rd ) be a measurable (F-non-anticipative) integrably bounded convex-valued set-valued process such that E(0, ·) < ∞. Then for every fixed 0 ≤ s < t < ∞ such that E(t, ·) < ∞ there exists the set H ⊂ [s, t] of the Lebesgue measure zero such that for every sequence (πr )r≥1 of partitions πr = {s = t0r < t1r < . . . < tlrr = t} of the interval [s, t], with {t0r , t1r , . . . , tlrr } ⊂ [s, t] \ H for every r ≥ 1, such that δr → 0 as t lr −1 r r → ∞, one has limr→∞ Eh( s (τ, ·)dτ, i=0

ti · (tir , ·)) = 0, where r r r r r δr = max0≤i≤lr −1 (ti+1 − ti ), and ti = ti+1 − ti for i = 0, 1, . . . , lr − 1, and every r ≥ 1. Proof Let 0 ≤ s < t < ∞ be such that E(t, ·) < ∞. By Theorem 4.1.11, there exists a sequence (j )∞ j =1 of measurable bounded convex-valued stochastic set-valued processes j : R+ ×  → Cl(Rd ) such t that limj →∞ E s h(j (τ, ·), (τ, ·))dτ = 0 and Ej (τ, ·) ≤ j for a.e. τ ∈ [s, t]. By Theorem 4.1.10, for every j ≥ 1 there exists a sequence j j + d (n )∞ n=1 of continuous bounded processes n : R ×  → Cl(R ) such that j j limn→∞ h(n (τ, ω), j (τ, ω)) = 0 and n (τ, ω) ≤ j for a.e. (τ, ω) ∈ [s, t] × t j . In particular, hence it follows that limn→∞ s Eh(n (τ, ·), j (τ, ·))dτ = 0 for every j ≥ 1. Assume, taking appropriate subsequences if needed, that j limj →∞ Eh(j (τ, ·), (τ, ·)) = 0 and limn→∞ Eh(n (τ, ·), j (τ, ·)) = 0 for every j ≥ 1, and a.e. τ ∈ [s, t]. Let Hj ⊂ [s, t] be for every j ≥ 1, the set of the Lebesgue measure zero such that limj →∞ Eh(j (τ, ·), (τ, ·))dτ = 0, and  j limn→∞ Eh(n (τ, ω), j (τ, ω)) = 0 for every τ ∈ [s, t]\Hj . Put H = j ≥1 Hj , and let (πr )r≥1 be a sequence of partitions πr = {s = t0r < t1r < . . . < tlrr = t}

132

4 Aumann Stochastic Integrals

of the interval [s, t], with {t0r , t1r , . . . , tlrr } ⊂ [s, t] \ H for every r ≥ 1, such that r − t r ) → 0 as r → ∞. For every i, n, r ≥ 1 one gets max0≤i≤lr −1 (ti+1 i



t

Eh

(τ, ·)dτ,

s





l r −1

tir

· (tir , ·)

 ≤ Eh



t

t

j (τ, ·)dτ,

s

j

n (τ, ·)dτ



 + Eh

s

t

t

(τ, ·)dτ,

s

i=0

Eh

t

j (τ, ·)dτ +

s

j

n (τ, ·)dτ,

s

l r −1

j

tir · n (tir , ·) +

i=0



l

l l l r −1 r −1 r −1 r −1 j r r r r r r r r

ti ·n (ti , ·),

ti ·j (ti , ·) +Eh

ti ·j (ti , ·),

ti ·(ti , ·) . Eh i=0

i=0

i=0

i=0

But,



t

Eh



s



t

Eh

Eh

t







j

n (τ, ·)dτ

l r −1

j · n (tir , ·)

tir

j

tir · n (tir , ·),

i=0

h((τ, ·), j (τ, ·))dτ, j

h(j (τ, ·), n (τ, ·))dτ,

s

≤

i=0

l r −1

t

≤E

s

j n (τ, ·)dτ,

t

≤E s

t

j (τ, ·)dτ,

s

Eh



j (τ, ·)dτ

s

s





t

(τ, ·)dτ,

l r −1  t r i+1 i=0

l r −1

tir

j

j

Eh(n (τ, ·), n (tir , ·))dτ,

l r −1 j

tir · j (tir , ·) ≤

tir Eh(n (tir , ·), j (tir , ·))

i=0

i=0

and Eh

l r −1

tir

· j (tir , ·),

l r −1

i=0



tir

· (tir , ·)

≤

i=0

l r −1

tir Eh(j (tir , ·), (tir , ·)).

i=0

Therefore, by properties of the sequence (πr )∞ r=1 of partitions the interval [s, t], and j ∞ ∞ sequences (j )j =1 and (n )n=1 of set-valued stochastic processes, for every r ≥ 1 one obtains



t

Eh s

(τ, ·)dτ,

l r −1 i=0



tir

· (tir , ·)

≤

(4.4.5)

4.4 Approximation of Aumann Stochastic Integrals





limj →∞ limn→∞ Eh

t

133

j n (τ, ·)dτ,

s

l r −1

tir

j · n (tir , ·)

 .

i=0

But, for every fixed j ≥ 1 one has



t

Eh

j n (τ, ·)dτ,

l r −1

s l r −1  t r i+1 tir

i=0

j



tir

j · n (tir , ·)

≤

i=0

j

Eh(n (τ, ·), n (tir , ·))dτ ≤ t

j

j

E[h(n (u, ·)n (v, ·))].

sup u,v∈[s,t],|u−v|≤δr

For every fixed j ≥ 1 and every n ≥ 1 there exists a positive integer rn ≥ 1 such that δn ≤ 1/(j n2 ) for every r ≥ rn . Therefore, for every fixed j ≥ 1, every n ≥ 1 and r ≥ rn one gets



t

Eh

j n (τ, ·)dτ,

s

l r −1

j

tir ·n (tir , ·)

≤t

sup

E[h(n (u, ·)n (v, ·))].

u,v∈[s,t],|u−v|≤1/(j n2 )

i=0

It is clear that positive integers rn ≥ 1 can be selected in such a way that a sequence (rn )∞ n=1 is increasing. Similarly to the proof of Theorem 4.4.2, for every |u − v| ≤ 1/(j n2 ), for every fixed j ≥ 1 and n ≥ 1 one gets j

j

E[h(n (u, ·)n (v, ·))] ≤   n

(v−1/n)∨0 (u−1/n)∨0

    Ej (τ, ·)dτ  + n

which implies that limn→∞ Eh( every fixed j ≥ 1. Thus,

t s

j

v u

  2 Ej (τ, ·)dτ  ≤ 2n · |u − v| · j ≤ , n

n (τ, ·)dτ,

lrn −1 i=0

j

tirn · n (tirn , ·)) = 0 for

 t  lrn −1  j j rn rn limj →∞ limn→∞ Eh n (τ, ·)dτ,

ti · n (ti , ·) = 0. s

i=0

t lrn −1 rn Hence, by (4.4.5) it follows limn→∞ Eh( s (τ, ·)dτ, i=0

ti ·(tirn , ·))=0. Let us note that the above procedure can be applied starting with an arbitrary taken increasing sequence (rj )∞ j =1 of positive integers. Then for we can select its subsequence (rn )∞ every such sequence (rj )∞ j =1 n=1 such that t lrn −1 rn limn→∞ Eh( s τ dτ, i=0 ti · t rn ) = 0. Thus, for every sequence (πr )r≥1 i of partitions πr = {s = t0r < t1r < . . . < tlrr = t} of the interval [s, t], with

134

4 Aumann Stochastic Integrals

r − tr) → 0 {t0r , t1r , . . . , tlrr } ⊂ [s, t] \ H for every r ≥ 1, such that max0≤i≤lr −1 (ti+1 i t lr −1 r as r → ∞, one has limr→∞ Eh( s τ dτ, i=0 ti · tir ) = 0.  

From Theorem 4.4.4 the following result follows. Theorem 4.4.5 If  : R+ ×  → Cl(Rd ) is a measurable (F-non-anticipative) integrably bounded convex-valued set-valued stochastic process, t t  t then for every 0 ≤ t s < t < ∞ one has () s τ dτ = s τ dτ ((F ) s τ dτ = s τ dτ ) a.s. Proof Let 0 ≤ s < t < ∞ be fixed. By relationships between Aumann and t Lebesgue set-valued stochastic integrals presented above, we have () s τ dτ ⊂ t ε ∞ s τ dτ a.s. By Corollary 4.1.7, for every ε > 0 there exists a sequence (n )n=1 of ε + continuous bounded convex-valued stochastic set-valued processes n : R × → t Cl(Rd ) such that limn→∞ E s h(εn (τ, ·), (τ, ·))dτ ≤ ε. By Theorem 4.1.7, it follows that

 t  t ¯ τ dτ, () τ dτ ≤ (4.4.6) Eh s

E h¯





t

t

τ dτ, s

s

s

εn (τ, ·)dτ

+ E h¯



 t εn (τ, ·)dτ, () εn (τ, ·)dτ +

t

s

s



 t  t E h¯ () εn (τ, ·)dτ, () (τ, ·)dτ ≤ s



t

E s

¯ h((τ, ·), εn (τ, ·))dτ + E

s



t s

¯ εn (τ, ·), (τ, ·))dτ, h(

t t because of Theorem 4.4.3, we have s εn (τ, ·)dτ = () s εn (τ, ·)dτ . Hence, by   ¯ t τ dτ, () t τ dτ ) ≤ the property of the sequence (εn )∞ n=1 , it follows that E h( s s  t τ dτ ) ≤ ¯ t τ dτ, (A) 2ε for every ε > 0 because by (4.4.6) one has E h( s s t t t 2 limn→∞ E s h(εn (τ, ·), (τ, ·))dτ . Then s τ dτ ⊂ () s τ dτ a.s. Theret t   fore, () s τ dτ = s τ dτ a.s. t Remark 4.4.3 If the assumptions of Theorem 4.4.5 are satisfied, then () s τ dτ = t t t t t (A) s τ dτ = s τ dτ , and (F ) s τ dτ = (AF ) s τ dτ = s τ dτ ) a.s. for every 0 ≤ s < t < ∞, respectively.  

4.5 Selection Theorems for Aumann Stochastic Integrals We consider now existence of some selectors of Aumann stochastic integrals. Similar problems have been considered in the author monograph [48] (see [48], Th. 1.4 and Th. 1.5 of Chap. 3) for set-valued stochastic functional integrals.

4.5 Selection Theorems for Aumann Stochastic Integrals

135

Lemma 4.5.1 If  : R+ ×  → Cl(Rd ) is a measurable p-integrably bounded convex-valued  tset-valued stochastic process, then for every fixed  t0 ≤ s < t < ∞ and u ∈ SF [ s τ dτ ] there exists ϕ ∈ S() such that u(ω) = s ϕτ (ω)dτ for a.e. ω ∈ . Proof By Theorem 4.4.5, the definition of the Lebesgue set-valued stochastic t integral () s τ dτ , and the decomposability of the set Js,t (S()), the following t equalities SF [ s τ dτ ] = decF Js,t (S()) = cl[Js,t (S())] follow. Hence, similarly to the proof of Corollary 4.4.3, there exists ϕ ∈ S() such that u(ω) = t ϕ (ω)dτ for a.e. ω ∈ .   s τ t The above result does not hold true for the set-valued stochastic integral s τ dτ defined for an F-non-anticipative multifunction and an Ft -measurable selector of t t  dτ , i.e., in the general case for u ∈ SFt [ s τ dτ ] there does not exist an τ s t F-non-anticipative selector ϕ of  such that u(ω) = s ϕτ (ω)dτ for a.e. ω ∈ , because Js,t (SF ()) is not decomposable subset of the space L(, Ft , Rd ). In such case we have the following result. Lemma 4.5.2 If  : R+ ×  → Cl(Rd ) is an F-non-anticipative integrably bounded convex-valued set-valued stochastic process, then for every fixed 0 ≤ t s < t < ∞ and u ∈ SFt [ s τ dτ ], there exists ϕ ∈ decβt ⊗Ft (SF ()) such that t u(ω) = s ϕτ (ω)dτ for a.e. ω ∈ , where βt denotes the Borel σ -algebra on [s, t]. t Proof By Theorem 4.4.5, for every u ∈ SFt ( s τ dτ ) one has u ∈ t t SFt ((F ) s τ dτ ) = decFt (Js,t (SF (F )). Then for every u ∈ SFt ( s τ dτ ), there n p exists a sequence (un )∞ n=1 ⊂ decFt [Js,t (SF ())] such that E|u − u| → 0 as n n → ∞. Therefore, for every n ≥ 1 there exists an Ft -measurable partition (Ank )N k=1  N N n n of  and a family (ϕkn )k=1 ⊂ SF () such that un = Js,t ( k=1 1[s,t]×Ank ϕkn ). But, N n n n k=1 1[s,t]×Ak ϕk ∈ decβt ⊗Ft [SF ()], and it does not belong to SF (). It is clear that decβt ⊗Ft [SF ()] is integrably bounded. Then it is a relatively weakly compact subset of the space L1 (R+ × , βt ⊗ Ft , Rd ). Therefore, taking an appropriate  n n n ∞ subsequence if needed, we infer that a sequence ( N k=1 1[s,t]×Ak ϕk )n=1 weakly converges to ϕ ∈ clw {decβt ⊗Ft [SF ()]}. By Theorem 3.3.3 of Chapter 3, one has decβt ⊗Ft [SF ()] = decβt ⊗Ft [co SF ()] = co{decβt ⊗Ft [SF ()]}. Therefore, decβt ⊗Ft [SF ()] is a convex subset of the space L1 (R+ × , βt ⊗ Ft , Rd ). Then ϕ ∈ clw {decβt ⊗Ft [SF ()]} = decβt ⊗Ft [SF ()]. Hence, by Dunford–Schwarz  n n n theorem it follows that Js,t ( N k=1 1[s,t]×Ak ϕk )  Js,t (ϕ) as n → ∞. But, t Nn E|Js,t ( k=1 1[s,t]×Ank ϕkn ) − u|p → 0 as n → ∞. Therefore, u = s ϕτ dτ a.s.   We can now prove the following selection theorems. Theorem 4.5.1 Let F : R+ ×  → Cl(Rd ) be a measurable integrably bounded convex-valued set-valued stochastic process, T > 0, and let x = (xt )0≤t≤T be a d-dimensional continuous stochastic process defined on the probability space

136

4 Aumann Stochastic Integrals

t (, F, P ). Then xt − xs ∈ s Fτ dτ a.s. for every 0 ≤ s < t ≤ T if and only t if there exists f ∈ S(F ) such that xt = x0 + 0 fτ dτ a.s. for every t ∈ [0, T ]. Proof Let ε > 0 and δε > 0 be such that sup0≤t≤T E|xt+δε − xt | ≤ ε/4, and  t+δ sup0≤t≤T E t ε Fτ dτ ≤ ε/4. Let πε = {0 < δε < 2δε < . . . < κδε = T } be a t partition of the interval [0, T ]. If xt − xs ∈ s Fτ dτ a.s. for every 0 ≤ s < t ≤ T , then by Lemma 4.5.1, for every i = 0, 1, . . . , κ − 1 and τi = iδε there exists τ ϕ i ∈ S(F ) such that xτi+1 − xτi = τii+1 ϕτi dτ . Let f ε = 1[0,τ1 ] ϕ 1 + 1(τ1 ,τ2 ] ϕ 2 + . . . + 1(τκ−1 ,T ] ϕ κ−1 . For fixed 0 ≤ s < t ≤ T there exist 1 ≤ p < q ≤ κ be such t ε that τp−1 ≤ s < t − xs ) − s fτ dτ | ≤  τpτp ε< . . . < τq−1 < t ≤ τq . Then  τp+1 E|(x E|(xp − xs ) − s fτ dτ | + E|(xτp+1 − xτp ) − τp fτε dτ | + . . . + E|(xτq−1 −  τq−1 ε  τt xτq−2 ) − τq−2 fτ dτ | + E|(xτq−1 − xt ) − τq−1 fτε dτ |, which implies that E|(xt − t ε  τp t xs ) − s fτ dτ | ≤ E|xp − xs | + s Fτ dτ | + E|xτq−1 − xt | + τq−1 Fτ dτ | ≤ ε. t Hence it follows that limε→0 E|(xt − xs ) − s fτε dτ | = 0. By weak compactness of S(F ), it follows that a family {f ε : ε > 0} is a relatively weakly compact subset of S(F ). Then there exists a sequence (εk )∞ k=1 of positive numbers convergent to 0 such that (f εk )∞ converges weakly to f ∈ S(F ). In particular, we also have  t k=1 t εk E|(xt −xs )− s fτ dτ | → 0 as k → ∞, because limε→0 E|(xt −xs )− s fτε dτ | = t 0. Therefore, xt − xs = s fτ dτ a.s. Conversely, if there exists f ∈ S(F ) such that t xt − xs = s fτ dτ a.s. for every 0 ≤ s < t ≤ T , then xt − xs ∈ Js,t (S(F )) = t t SF ( s Fτ dτ ) for every 0 ≤ s < t ≤ T , which implies that xt − xs ∈ s Fτ dτ a.s. for every 0 ≤ s < t ≤ T .   Theorem 4.5.2 Let F : R+ ×  → Cl(Rd ) be an F-non-anticipative integrably bounded convex-valued set-valued stochastic process, and let x = (xt )0≤t≤T be a d-dimensional continuous F-adapted stochastic process on a filtered probability t space (, F, F, P ). If xt − xs ∈ s Fτ dτ a.s. for every 0 ≤ s < t ≤ T , then there t exists f ∈ decβT ⊗FT [SF (F )] such that xt = x0 + 0 fτ dτ a.s. for every t ∈ [0, T ]. t Proof By xt − xs ∈ s Fτ dτ a.s. for every 0 ≤ s < t ≤ T it follows that xt − xs ∈ t SFt ( s Fτ dτ ) for every 0 ≤ s < t ≤ T . Let (πn )∞ n=1 be the sequence of partitions of the interval [0, T ] of the form πn = {0 ≤ τ1n < τ2n < . . . < τκnn −1 < τκnn with τin = iδn , δn = T /2n , and κn = 2n . By virtue of Lemma 4.5.2, for every n ≥ 1 and i = 1, 2, . . . , κn , there exists ϕin ∈ decβτ n ⊗Fτ n [SF ()] such that xτin (ω) − i i  τin n n (ω) = ϕ (τ, ω)dτ for a.e. ω ∈ , where β xτi−1 n τin the Borel σ -algebra on the i τ i−1

n , τ n ]. Let f n = 1 n n n interval [τi−1 [0,τ1n ] ϕ1 + 1(τ1n ,τ2n ] ϕ2 + . . . + 1τκn −1 ,T ] ϕκn −1 for every i n

n ≥ 1. It is clear that f n ∈ decβT ⊗FT [S(F )] for every n ≥ 1. Furthermore, for every n ] and t ∈ [τ n , τ n ] with 0 ≤ p ≤ q ≤ κ 0 ≤ s < t ≤ T such that s ∈ [τpn , τp+1 n q−1 q one has    t   n  fτ dτ  = E (xt − xs ) − s

4.6 Indefinite Aumann Stochastic Integrals

137

  n ) + (xτ n n ) + . . . + (xτ n n ) + (xτ n E (xt − xτq−1 − xτq−2 − xτp+1 − xs )− q−1 p+2 p+1 



t n τq−1

fτn dτ −

n τq−1 n τq−2

   4 max E 

v u

 fτn dτ − . . . −

n τp+1 n τp+2

 fτn dτ −

s

n τp+1

  fτn dτ  ≤

   F (τ, ·)dτ  : u, v ∈ [0, T ], |u − v| ≤ δn .

t Therefore, limn→∞ E|(xt − xs ) − s fτn dτ | = 0. By virtue of Lemma 4.5.2, there n ∞ such that exists f ∈ decβT ⊗FT [S(F )], and a subsequence (f nk )∞ k=1 of (fn=1 )  t nk t t nk as k → ∞. But limk→∞ E|(xt − xs ) − s fτ dτ |p = 0. s fτ dτ  s fτ dτ t t Therefore, xt −xs = s fτ dτ a.s. for every 0 ≤ s < t ≤ T . Thus xt = x0 + 0 fτ dτ a.s. for every t ∈ [0, T ].  

4.6 Indefinite Aumann Stochastic Integrals Given a measurable integrably bounded set-valued stochastic process F : R+ × t  → Cl(Rd ) we can define a set-valued stochastic process ( 0 Fτ dτ )t≥0 , called the indefinite Aumann stochastic integral. We shall show that it is continuous. We begin with the following lemmas. Lemma 4.6.1 Let X and Y be linear spaces and T : X → Y be a linear mapping. Then for every nonempty sets H, K ⊂ X one has T (H + K) = T (H) + T (K). Proof For every u ∈ T (H + K) there are f ∈ H and g ∈ K such that u = T (f +g) = T (f )+T (g). Then for every u ∈ T (H+K) one has u ∈ T (H)+T (K). Thus, T (H + K) ⊂ T (H) + T (K). On the other hand, for every u ∈ T (K) + T (H) there are f ∈ K and g ∈ H such that u = T (f ) + T (g) = T (f + g) ∈ T (K + H). Then T (K) + T (H) ⊂ T (K + H).   Lemma 4.6.2 For every convex-valued measurable integrably bounded set-valued stochastic process F : R+ ×  → Rd , every 0 ≤ s < t < ∞ and δ > 0 such that t  s+δ t s + δ < t one has s Fτ dτ = s Fτ dτ + s+δ Fτ dτ a.s. t t Proof By virtue of Theorem 4.4.5, one has () 0 Fτ dτ = 0 Fτ dτ a.s. By the t definition of the Lebesgue set-valued stochastic integral () 0 Fτ dτ , it follows that t SF ( s Fτ dτ ) = decF [Js,t (S(F ))]. By Lemmas 4.6.1 and 3.3.4 of Chapter 3, it follows that Js,t (S(F )) = Js,s+δ (S(F )) + Js+δ,t (S(F )), which by Corollary 1.2.4 of Chapter 1 implies that decF [Js,t (S(F ))] = decF [Js,s+δ (S(F ))] + decF [Js+δ,t (S(F ))], because decF [Js,s+δ (S(F ))] and decF [Js+δ,t (S(F ))] are t  s+δ t weakly compact. Then SF ( s Fτ dτ ) = SF ( s Fτ dτ ) + SF ( s+δ Fτ dτ ). But,

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4 Aumann Stochastic Integrals

 s+δ t by Theorem 2.3.6 of Chapter 2, one has SF ( s Fτ dτ ) + SF ( s+δ Fτ dτ ) =  s+δ t t t SF ( s Fτ dτ + s+δ Fτ dτ ), because s+δ Fτ dτ and s+δ Fτ dτ are compact  s+δ t convex subsets of Rd a.s., and SF ( s Fτ dτ ) and SF ( s+δ Fτ dτ ) are convex weakly compact subsets of the space L1 (, F, Rd ). Therefore, by Corollary 2.3.1  s+δ t t   of Chapter 2, one gets s Fτ dτ = s Fτ dτ + s+δ Fτ dτ a.s. Theorem 4.6.1 For every measurable integrably bounded set-valued stochastic process F : R+ ×  → Cl(Rd ) the indefinite Aumann stochastic integral t ( 0 Fτ dτ )t≥0 is a continuous set-valued stochastic process. Proof Let t > 0 and δ > 0. By (iv) of Theorem 4.1.7, we can assume that F is convex-valued. By virtue of Lemma 4.6.2, and properties of the Hausdorff metric, one gets

 t+δ

 t  t  t+δ  t ¯h ¯ Fτ dτ, Fτ dτ = h Fτ dτ + Fτ dτ, Fτ dτ 0

0

) ) ≤) )

t

0 t+δ

) ) Fτ dτ ) ),

t

0

 t+δ  t+δ where  t Fτ dτ  = sup{|x| : x ∈ t Fτ dτ }. By integrable boundedness of F there exists an integrable function λ : R+ ×  → R+ such that Fτ (ω) ≤  t+δ λτ (ω) for a.e. (τ, ω) ∈ R+ × . Then for a.e. ω ∈  we have  t Fτ (ω)dτ  ≤  t+δ t  t+δ  t+δ ¯ Fτ (ω)dτ ≤ t λτ (ω)dτ . Therefore, limδ→0 h( Fτ dτ, 0 Fτ dτ ) = 0 t 0 t a.s. Then a set-valued process ( 0 Fτ dτ )t≥0 is h-u.s.c., which by [27, Prop.2.68 of t Chap. 1] implies that it is u.s.c., because 0 Fτ dτ is a compact convex subset of the   ¯ t Fτ dτ, t+δ Fτ dτ ) = 0, which space Rd . In a similar way, we obtain limδ→0 h( 0 0 t   implies that a set-valued process ( 0 Fτ dτ )t≥0 is l.s.c. Remark 4.6.1 In a similar way, we can verify that the set-valued process t ( 0 Fτ dτ )t≥0 is continuous for every F-non-anticipative integrably bounded setvalued stochastic process F : R+ ×  → Rd .  

4.7 Notes and Remarks The results of this chapter are based on the author papers [53, 56] and [58]. The definition and basic properties of Aumann stochastic functional integrals have been presented first in the author papers [39] and [40]. Later on, they were investigated in the monograph [48]. The results dealing with approximations of stochastic Aumann and Lebesgue set-valued integrals and relations between these set-valued integrals come from the paper [53]. Selection theorems of Aumann  t stochastic integrals come from the paper [58]. Set-valued stochastic integral (A) s Fτ dτ has been considered first by I. Li and S. Li in [63] and [64]. These integrals are called there as “set-valued

4.7 Notes and Remarks

139

stochastic Lebesgue integral” and “Aumann type set-valued stochastic  t Lebesgue integral,” respectively. The Lebesgue set-valued stochastic integral () s Fτ dτ has been defined and investigated first in the paper [56]. It was defined  t by the procedure presented in the paper [33]. The Aumann stochastic integrals s Fτ dτ depending on a random parameter were considered first by A. Fryszkowski in [17]. A selection problem for such defined Aumann integrals was considered by A. Fryszkowski and Ch. Hess in [17] and [22], respectively. It was shown in the author papers [53] that the above-defined Aumann stochastic integrals depending on a random parameter and the Lebesgue set-valued stochastic integrals are equal a.s. These equalities have important consequences. Among others they imply that properties of all the abovedefined set-valued stochastic integrals follow immediately from properties of the classical Aumann integrals depending on random parameters. The above equalities follow from approximation theorems presented in the author paper [53]. These theorems can be also applied to the theory of stochastic differential inclusions to constructions of their approximation solutions. Results presented in [63] and [64] have been extended by J. Zhang and J. Qi (see [99]) on the case of set-valued stochastic integrals defined with respect to finite variation processes. The classical Aumann integral, defined by R. Aumann in the paper [5], has been extended to the case of set-valued Bochner and Lebesgue–Stieltjes integrals (see [4, 24, 60, 65, 95] and [96]), respectively. The integrals, depending on random parameters, can be applied to the definition of stochastic fuzzy integrals, considered among others in the paper [68]. Theorem 4.1.7 is taken from F. Hiai and H. Umegaki [24]. Lemma 4.1.4 comes from J. Distel and J. Uhl [13], whereas Theorems 4.1.8 and 4.1.9 are selected from the author monograph [38].

Chapter 5

Itô Set-Valued Integrals

In this chapter we present the definition and properties of Itô set-valued integrals of square integrable non-anticipative matrix-valued stochastic processes. We begin with the definition and properties of Itô set-valued functional integrals of subsets of the space L2 (R+ × , F , Rd×m ).

5.1 Itô Set-Valued Functional Integrals Similarly as above let us assume that we are given a complete filtered probability space PF = (, F, F, P ) with a filtration F = (Ft )t≥0 satisfying the usual conditions. Furthermore, assume we are given an m-dimensional F-Brownian motion B = (Bt )t≥0 defined on PF . Throughout this section we shall consider the linear mapping Js,t , defined for fixed 0 ≤ s < t < ∞ on the space L2 (R+ × , F , Rd×m ) with values at L2 (, Ft , P ), by setting Js,t (ψ)(ω) = t ( s ψτ dBτ )(ω) for a.e ω ∈  and ψ ∈ L2 (R+ × , F , Rd×m ). From properties of Itô integrals, it follows that Js,t : L2 (R+ × , F , Rd×m ) → L2 (, Ft , P ) is a linear isometry. Given a nonempty set K ⊂ L2 (R+ ×, F , Rd×m ) the Itô set-valued functional integral Js,t (K) of K over the interval [s, t] is defined by setting Js,t (K) = {Js,t (g) : g ∈ K}. It is clear that for every closed set K ⊂ L2 ([s, t] × , F , Rd×m ), the Itô set-valued functional integral Js,t (K), is a closed subset of the space L2 (, Ft , P ). Indeed, for every u ∈ cl[Js,t (K)] there is a sequence n (g n )∞ in particular, it n=1 of K such that | Js,t (g ) − u | → 0 as n → ∞. Hence t n n m follows that | Js,t (g ) − Js,t (g ) | → 0 as n, m → ∞. But, E s |gτ − gτm |2 dτ = | Js,t (g n ) − Js,t (g m ) | 2 . Therefore, (g n )∞ n=1 is the Cauchy sequence of the Banach space L2 ([s, t] × , F , Rd×m ), which by the closedness of the set K, implies that there exists g ∈ K such that | Js,t (g n ) − Js,t (g) | → 0 as n → ∞. Thus, © Springer Nature Switzerland AG 2020 M. Kisielewicz, Set-Valued Stochastic Integrals and Applications, Springer Optimization and Its Applications 157, https://doi.org/10.1007/978-3-030-40329-4_5

141

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5 Itô Set-Valued Integrals

u = Js,t (g) ∈ Js,t (K) for every u ∈ cl[Js,t (K)]. Then cl[Js,t (K)] ⊂ Js,t (K). From properties of the Itô integral it follows that for every nonempty set K ⊂ L2 (R+ × , F , Rd×m ), one has E[Js,t (K)] = {0}. Indeed, by the definition of the Itô integral and the definition of the mean value of set-valued random variable, we have E[Js,t (K)] = {E[Js,t (g)] : g ∈ K} = {0}. The next question, concerning the Itô set-valued functional integral Js,t (K), deals with its decomposability. Unfortunately, in the general case the answer to such question, even for simplest sets K ⊂ L2 (R+ × , F , Rd×m ), is not positive. It follows from the following simple example. Example 5.1.1 Let K be the set of all F-non-anticipative selectors of the set-valued mapping , defined by (t, ω) = : [0, 1] for t ∈ [0, 1] and ω ∈ , and let J (K) = : J0,1 (K). Suppose J (K) is decomposable. Then for every A ∈ F1 , and every 1 u, v ∈ J (K), one has 1A u + 1A∼ v ∈ J (K). Take in particular, u = 0 1dBt = B1 , 1 v = 0 0dBt = 0 a.s. and let A = {ω ∈  : B1 ≥ ε} for ε > 0. We have A ∈ F1 . On the other hand, by the definition of J (K) there is ψ ∈ K such that 1 1 1A u + 1A∼ v =  1A · B1 = 0 ψt dBt . Then E(1A · B1 ) = E[ 0 ψt dBt ] = 0, and E(1A · B1 ) = {B1 ≥ε} B1 dP ≥ ε · P (B1 ≥ ε) > 0. Contradiction.   It can be proved that it does not exist a set K ⊂ L2 (R+ × , F , Rd×m ), containing a more than one element and such that its Itô set-valued functional integral Js,t (K) is decomposable. We have the following properties of Itô set-valued functional integrals of subsets of the space L2 (R+ × , F , Rd×m ). Lemma 5.1.1 For every 0 ≤ s < t < ∞, and nonempty sets G, H ⊂ L2 (R+ × , F , Rd×m ), one has Js,t (K + H) = Js,t (K) + Js,t (H) , Js,t [cl(G)] = cl[Js,t (G)] , Js,t (co(G)) = co Js,t (G) , dec{Js,t [co(G)]} = co[dec Js,t (G)] , if (, F, P ) is separable probability space, then there exists a sequence n (g n )∞ n=1 ⊂ G such that cl Js,t (G) = cl{Js,t (g ) : n ≥ 1} , (vi) Js,t (K) is decomposable if and only if K is a singleton , (vii) for every 0 ≤ s ≤ α ≤ β ≤ t one has Js,t (1[α,β] · G) = Jα,β (G). (i) (ii) (iii) (iv) (v)

Proof (i) The result follows immediately from Lemma 4.6.1 of Chapter 4 and the linearity of Js,t . (ii) By continuity of the mapping Js,t , one has Js,t [cl(G)] ⊂ cl[Js,t (G)]. For every u ∈ cl[Js,t (G)] and every sequence (g n )∞ n=1 ⊂ G such that n )−J (g m )|2 → 0 as m, n → ∞. E|u−Js,t (g n )|2 → 0, we have E|Js,t (g s,t t But , E|Js,t (g n ) − Js,t (g m )|2 = E s |gτn − gτm |2 dτ for every m, n ≥ 2 1. Then (g n )∞ n=1 is a Cauchy sequence of the Banach space L ([s, t] × d×m 2 d×m , F , R ). Thus, there exists g ∈ L ([s, t] × , F , R ) such that

5.1 Itô Set-Valued Functional Integrals

(iii)

(iv)

(v)

(vi)

(vii)

143

t E s |gτn − gτ |2 dτ → 0 as n → ∞, which implies that g ∈ cl(G) and E|Js,t (g) − Js,t (g n )|2 → 0 as n → ∞. Then Js,t (g) ∈ Js,t [cl(G)], which implies that cl[Js,t (G)] ⊂ Js,t [cl(G)]. By the linearity of the mapping Js,t we have Js,t (co G) = co Js,t (G), which by Corollary 1.2.2 of Chapter 1, implies that cl[Js,t (co G)] = co Js,t (G). Hence, by (ii) it follows that Js,t (co(G)) = co [Js,t (G)], because cl[Js,t (co G)] = Js,t (co G). It is clear that Js,t [co(G)]} ⊂ co[dec Js,t (G)], because Js,t (co(G)) = co Js,t (G) and Js,t (G) ⊂ dec Js,t (G). Let us observe that by Theorem 3.3.3 of Chapter 3, co[dec Js,t (G)] is a decomposable subset of the space L2 (, Ft , Rd ). Therefore, dec{Js,t [co(G)]} ⊂ co[dec Js,t (G)]. On the other hand, we have dec Js,t (G) ⊂ dec{Js,t [co(G)]}. By virtue of Lemma 3.3.3 of Chapter 3, dec{Js,t [co(G)]} is a convex subset of the space L2 (, Ft , Rd ). Then co[dec Js,t (G)] ⊂ dec {Js,t [co(G)]}. Thus, dec{Js,t [co(G)]} = co[dec Js,t (G)]. By the separability of the space (, F, P ) the space L2 (R+ × , F , Rd×m ) is separable, because it is a closed subset of the separable Banach space L2 (R+ × , β ⊗ F, Rd×m ). Then the set G ⊂ L2 ([s, t] × , F , Rd×m ) with its induced topology is the separable subspace of L2 (R+ × , F , Rd×m ). Thus, there is a sequence (g n )∞ n=1 ⊂ G such that G = clI {g n : n ≥ 1}, where clI denotes the closure in the induced topology of G. But cl{g n : n ≥ 1} ⊂ cl[G] and clI {g n : n ≥ 1} = G ∩cl{g n : n ≥ 1}. Then cl G = cl[clI {g n : n ≥ 1}] = cl[G ∩ cl{g n : n ≥ 1}] ⊂ cl G ∩ cl{g n : n ≥ 1} = cl{g n : n ≥ 1}, which by (ii) implies that cl Js,t (G) = cl{Js,t (g n ) : n ≥ 1}. It is clear that if Js,t (K) is a singleton, then it is decomposable. Suppose dec{Js,t (K)} = Js,t (K). Then for every A ∈ Ft , and every u, v ∈ Js,t (K), one has 1A u + 1\A v ∈ Js,t (K). But 1A u + 1\A v = 1A (u − v) + v, E[v] ∈ E{Js,t (K)} = {0} and E[1A u + 1\A v] ∈ E[Js,t (K)] = {0}. Therefore, for every A ∈ Ft , and u, v ∈ Js,t (K) one has E[1A Js,t (ϕ − ψ)] ∈ E{Js,t (K)} = {0}, where u = Js,t (ϕ) and v = Js,t (ψ). Then Js,t (ϕ − ψ) = 0, because Js,t (ϕ − ψ) is  t Ft -measurable. Therefore, E|Js,t (ϕ − ψ)|2 = 0, which implies that E s |ϕτ − ψτ |2 dτ = 0. Thus, if Js,t (K) is decomposable, then for every ϕ, ψ ∈ K one has ϕ = ψ. Then K is a singleton. t If 0 ≤ s ≤ α ≤ β ≤ t, then Js,t (1[α,β] · G) = { s 1[α,β] gτ dBτ : g ∈ G} = β { α gτ dBτ : g ∈ G} = Jα,β (G).  

Lemma 5.1.2 Let (, F, P ) be a separable probability space, (X, ρ) a metric space and, let  : X  x → (x) ⊂ L2 (R+ × , F , Rd×m ) be l.s.c. set-valued mapping with nonempty closed values. Then there exists a sequence n 2 + d×m ) such that (g n )∞ n=1 , of continuous functions g : X → L (R × , F , R n n g (x) ∈ co (x) for n ≥ 1, and co (x) = cl{g (x) : n ≥ 1} for every x ∈ X.

144

5 Itô Set-Valued Integrals

Proof The result follows immediately from Remark 2.1.1 of Chapter 2, because the space L2 (R+ × , F , Rd×m ) is separable, and the set-valued mapping X  t , Rd×m ) is l.s.c. with nonempty closed convex x → co (x) ⊂ L2 ([s, t] × , F values.   Lemma 5.1.3 If G = {g n : n ≥ 1} ⊂ L2 (R+ × , F , Rd×m ), then decFt [Js,t (G)] = Lim decFt [Js,t (Gp )], where Gp = {g 1 , . . . , g p } for p ≥ 1. Proof Let us observe that cl(G) = Lim Gp . Indeed, we have Gp ⊂ Gp+1 for every p ≥ 1. Therefore, by (c) of Lemma 1.3.5 of Chapter 1, the Kuratowski limit Lim Gp exists. Furthermore, Gp ⊂ cl(G) for every p ≥ 1, which implies that Lim Gp ⊂  cl(G), because by (c) of Lemma 1.3.5 of Chapter 1, one has Lim Gp = cl{ p≥1 Gp }. On the other hand, for every g ∈ cl(G) there exists a n ∞ nk → g as k → ∞. For every subsequence (g nk )∞ k=1 of (g )n=1 such that g n k k ≥ 1 there is pk ≥ 1 such that g ∈ Gpk . Then by (a) of Lemma 1.3.4 of Chapter 1, one has g ∈ Li Gp = Lim Gp . Therefore, cl(G) ⊂ Lim Gp . Thus, cl(G) = Lim Gp . In a similar way we obtain cl[Js,t (G)] = Lim Js,t (Gp ), which implies that decFt Js,t (G) = decFt Js,t (Lim Gp ) = decFt [Lim Js,t (Gp )], because decFt Js,t (G) = decFt Js,t (cl(G)) = decFt Js,t (Lim Gp ). To conclude the proof, one has to verify that decFt [Lim Js,t (Gp )] = Lim decFt [Js,t (Gp )]. It is clear that for every p ≥ 1 we have decFt {Js,t (Gp )} ⊂ decFt {Lim Js,t (Gp )}. Then Lim decFt {Js,t (Gp )} ⊂ decFt {Lim Js,t (Gp )}. On the other hand, for every a ∈ decFt {Lim Js,t (Gp )} there exists a sequence (ar )∞ r=1 of 2 decFt {Lim Js,t (Gp )} = decFt {Js,t (Lim Gp )} convergent to u ∈∈ L (, F, Rd ) in the norm topology of L2 (, F, Rd ). For every r ≥ 1, there are an Ft r r Nr measurable partition (Ark )N k=1 of , and a family (uk )k=1 ⊂ Js,t (Lim Gp ) Nr r r such that ar = k=1 1Ak uk . For every r ≥ 1, and k = 1, . . . , Nr there k,r ∈ Gp for every p ≥ 1, and such exists a sequence (vpk,r )∞ p=1 such that vp

that Js,t (vpk,r ) → urk in the norm topology of L2 (, F, Rd ) as p → ∞. Nr Then for every r ≥ 1 we have 1Ark Js,t (vpk,r ) ∈ decFt {Js,t (Gp )} for Nr k=1 Nm k,r r r r p ≥ 1 and k=1 1Ak Js,t (vp ) → k=1 1Ak uk = ar as p → ∞. Therefore, ar ∈ Li[decFt {Js,t (Gp )}] = Lim[decFt {Js,t (Gp )}] for every r ≥ 1, which implies that ar ∈ Lim[decFt {Js,t (Gp )}], because decFt {Js,t (Gp )} ⊂ decFt {Js,t (Gp+1 )} for every p ≥ 1. Hence it follows that u ∈ Lim[decFt Js,t (Gp )] for every u ∈ decFt {Lim Js,t (Gp )}. Thus, decFt {Lim Js,t (Gp )} ⊂ Lim[decFt Js,t (Gp )],   which implies that decFt {Lim Js,t (Gp )} = Lim decFt {Js,t (Gp )}. Remark 5.1.1 In particular case, if K is a closed decomposable subset of L2 (R+ × , F , Rd×m ), then by Theorem 2.3.3 of Chapter 2, there is an F-non-anticipative multifunction F : R+ ×  → Cl(Rd×m ) such that K = SF (F ). In such case Js,t (K) = Js,t (SF (F )) and Js,t (SF (F )) is said to be the Itô set-valued functional integral of F . The properties of such set-valued functional integrals were considered in the monograph [48] (see [48], Th.1.1 of Chap. 3).  

5.1 Itô Set-Valued Functional Integrals

145

We shall now consider properties of Itô set-valued functional integrals Js,t (KF ) of the set KF = {1D : D ∈ F }, defined with respect to a one-dimensional FBrownian motion B = (Bt )t≥0 . We begin with the following result. Lemma 5.1.4 Let 0 ≤ s < t < ∞ be fixed, K a nonempty closed decomposable subset of the space L2 (R+ × , F , Rd ), and assume a number  t thatk there exist k (τ )|2 dτ ≥ δ δ > 0, and a sequence (fn )∞ ⊂ K such that inf |f (τ ) − f j j ≥ 1, and some k ∈ {1, 2, . . . , d}, where fn = (fn1 , . . . , fnd ) for n ≥ 1. Then sup{E|u|2 : u ∈ dec Js,t (K)} = ∞ for every 0 ≤ s < t < ∞. Proof Let Xnk = Js,t (fnk ) for every n ≥ 1, and k ∈ {1, 2, . . . , d}. One has E|Xik

− Xjk |2

2  t  t   k k   = E  [fi (τ ) − fj (τ )]dBτ  = E |fik (τ ) − fjk (τ )|2 dτ. s

s

Therefore, by Fatou’s lemma for every j < i, one gets

  t δ ≤ E inf |fik (τ ) − fjk (τ )|2 dτ ≤

(5.1.1)

j j ≥ 1. Hence, by Sudakov-SlepianFernique theorem it follows that k |}])2 . (E[max{Y1 , . . . , YN }])2 ≤ (E[max{|X1k |, . . . , |XN

(5.1.2)

By Fernique theorem, there exists the constant K > 0 such that K 2 ln(N ) ≤ (E[max{Y1 , . . . , YN }])2 for every N ≥ 1. Hence, by (5.1.2) and the H¨older inequality for every k ∈ {1, 2, . . . , d}, and N ≥ 1 it follows that

146

5 Itô Set-Valued Integrals k k 2 K 2 ln(N ) ≤ (E[max{|X1k |, . . . , |XN |}])2 ≤ E[max{|X1k |2 , . . . , |XN | }].

k |2 }]. But, E[max{|X |2 , . . . , Then K 2 ln(N ) ≤ max1≤k≤d E[max{|X1k |2 , . . . , |XN 1 k |2 }], where X = (X 1 , . . . , X d ) for |XN |2 }] ≥ max1≤k≤d E[max{|X1k |2 , . . . , |XN j j j j = 1, 2, . . . , N . Therefore, K 2 ln(N ) ≤ E[max{|X1 |2 , . . . , |XN |2 }] ≤ M()2 , where  = Js,t (K), M() = ∞ N =1 {max{|u1 |, . . . |uN |} : u1 , . . . , uN ∈ }, and M()2 = sup{ | u | 2 : u ∈ M()}. But, M()2 ≤ E[decFt Js,t (K)2 and E[decFt Js,t (K)2 = E[decFt Js,t (K)]2 = sup{E|u|2 : u ∈ decFt Js,t (K). Therefore, for every N ≥ 1, one has K 2 ln(N ) ≤ sup{E|u|2 : u ∈ decFt Js,t (K)}, which implies that sup{E|u|2 : u ∈ dec Js,t (K)} = ∞.  

We can now prove the following result. Theorem 5.1.1 Let K be a nonempty closed decomposable subset of the space L2 (R+ ×, F , Rd ), and assume that there exist f, g ∈ K and k ∈ {1, 2, . . . , d} t such that s |f k (τ ) − g k (τ )|2 dτ > 0 a.s. Then sup{E|u|2 : u ∈ decFt [Js,t (K)] = ∞. Proof Suppose that sup{E|u|2 : u ∈ dec Js,t (K)} < ∞, and let  :  → Cl(Rd ) be the set-valued Ft -measurable random variable such that SFt () = decFt Js,t (K). Such set-valued random variable exists by Theorem 2.3.3 of Chapter 2. By Remark 2.3.2 of Chapter 2, we have sup{E|u|2 : u ∈ dec Js,t (K)} = sup{E|u|2 : u ∈ dec Js,t (K)} = sup{E|u|2 : u ∈ SFt ()} = E[sup{|x|2 : x ∈ }]. Therefore, E[sup{|x|2 : x ∈ }] = sup{E|u|2 : u ∈ dec Js,t (K)} < ∞, which implies that sup{|x k | : x ∈ (ω)} < ∞ for a fixed k ∈ {1, . . . , d} and a.e. ω ∈ , where x = (x 1 , . . . , x d ) for x ∈ (ω). By the properties of the set K, we ensure that λ({τ ∈ [s, t] : f k (τ ) > g k (τ )}) > 0 or λ({τ ∈ [s, t] : f k (τ ) < g k (τ )}) > 0 for any k ∈ {1, . . . , d}, where λ denotes the real Lebesgue measure. Suppose that λ({τ ∈ [s, t] : f k (τ ) > g k (τ )}) > 0. Without loss of the generality, we may assume that f k (τ ) > g k (τ ) for τ ∈ [s, t]. Otherwise, we can consider a set 1A · K with A = {τ ∈ [s, t] : f k (τ ) > g k (τ )} instead of K. By the equality sup{E|u|2 : u ∈ SFt ()} = E[sup{|x|2 : x ∈ }], and the assumption that sup{E|u|2 : u ∈ dec Js,t (K)} < ∞ it follows (see Corollary 2.3.4 of Chapter 2), that the set-valued random variable  is square integrably bounded. Then also the set-valued random variable  − Js,t (g) is integrably bounded, which implies that sup{E|u|2 : u ∈ dec Js,t (G)} < ∞, where G = K − g. Indeed, by the boundedness of , the set SFt () − Js,t (g) is a bounded subset of L2 (, Ft , Rd ). But, SFt () − Js,t (g) = dec Js,t (K) − Js,t (g) = dec Js,t (K − g) = dec Js,t (G). Then sup{E|u|2 : u ∈ dec Js,t (G)} < ∞. In a similar way we infer that sup{E|u|2 : u ∈ dec Js,t (H)} < ∞, where H = G −(f −g)/2. Let us note that −(f −g)/2, (f −g)/2 ∈ H. Indeed, we have 0 ∈ G and H¯ ({−(f −g)/2}, H) = H¯ ({−(f −g)/2}, G +{−(f −g)/2}) =

5.1 Itô Set-Valued Functional Integrals

147

¯ h({0}, G) = 0, where H¯ is the Hausdorff sub-distance on the space Cl(L2 ([s, t] × ¯ ¯ − g)/2}, H) = h({f/2} − {g/2}, {K − g − , F , Rd )). Similarly one gets h({(f ¯ ¯ f/2+g/2}) = h({f/2}+{−g/2}, {K−f/2}+{−g/2}) = h({f/2}, K+{−f/2}) = ¯ h({0}, K − f ) = 0, because 0 ∈ K − f . Now, by Lemma 1.4.1 of Chapter 1, for every k ∈ {1, 2, . . . , d}, there exists an orthogonal sequence (hkn )∞ n=1 of the space t k 2 t k 2 + L (R × , F , R) such that s (hn ) (τ )dτ = (1/2) s |f (τ ) − g k (τ )|2 dτ , and such that hkn (τ ) ∈ {−(f k (τ ) − g k (τ ))/2, (f k (τ ) − g k (τ ))/2} for n = 0, 1, 2, . . ., and a.e. τ ∈ [s, t]. By the construction of the sequence (hkn )∞ n=1 presented in the proof of Lemma 1.4.1 of Chapter 1, for every n = 0, 1, 2, . . . and a.e. τ ∈ [s, t], one has n

hkn (τ )

=

2 

(−1)r−1 1Drn (τ )

r=1

f k (τ ) − g k (τ ) , 2

where ({D1n , . . . , D2nn })∞ n=1 is the sequence of Borel measurable partitions of an interval [s, t], such that D10 = [s, t]. Let (un )∞ n=1 be the sequence of d-dimensional F-non-anticipative processes un = (u1n , . . . , ndn ) defined by n

ukn (τ, ω)

=

2  j =1

(−1)j −1 1Djn × (τ, ω)

f k (τ, ω) − g k (τ, ω) 2

for k = 1, 2, . . . , d and (τ, ω) ∈ [s, t] × . By the above definitions, and the ∞ k ∞ decomposability of the set H, we get (un )∞ n=1 ⊂ H and (un )n=1 = (hn )n=1 for k = 1, 2, . . . , d. Then the sequence (un )∞ and the set H satisfy the assumptions n=1 t of Lemma 5.1.4 with δ = (1/4) s |f k (τ ) − g k (τ )|2 dτ , because 

t s

2n

1 2

=



1 2n+1

s

t

|hki (τ ) − hkj (τ )|2 dτ =

n

|f (τ ) − g (τ )| dτ = k

n r=1 Dr



 |uki (τ ) − ukj (τ )|2 dτ

+

k

2

2  r=1

1 4



t s

1



2n+1

|f k (τ ) − g k (τ )|2 dτ ≥

t

|f k (τ ) − g k (τ )|2 dτ =

s

1 4



t

|f k (τ ) − g k (τ )|2 dτ

s

for every i > j ≥ 1. Therefore, sup{E|u|2 : u ∈ dec Js,t (H)} = ∞, which contradicts to the inequality sup{E|u|2 : u ∈ dec Js,t (H)} < ∞, stated above. Then sup{E|u|2 : u ∈ dec Js,t (K)} = ∞.   Remark 5.1.2 The result of Lemma 5.1.4 also holds true if K is a nonempty bounded closed decomposable subset of the space L2 (R+ × , F , Rd ).  

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5 Itô Set-Valued Integrals

Remark 5.1.3 By Theorem 5.1.1 we can construct a lot of examples of decomt , Rd ) possessing unbounded Itô posable subsets of the space L2 (R+ × , F set-valued functional integrals. For example, in the case of the real-valued F-nonanticipative processes, we can consider a set K = dec{f, g} for every f, g ∈ L2 (R+ × , F , R) such that ft (ω) < gt (ω) for e.e. (τ, ω) ∈ [s, t] × . Corollary 5.1.1 For every fixed 0 ≤ s < t < ∞ and KF = {1D : D ∈ F }, the set Js,t (KF ) is not square integrably bounded subset of the space L2 (, Ft , R). Proof It is clear that for every fixed 0 ≤ s < t < ∞, KF is a bounded decomposable subset of the space L2 ([s, t] × , R). To see that Js,t (KF ) is not t denotes square integrably bounded, let us note that KF = dec t {0, 1}, where F F the σ -algebra of F-non-anticipative subsets of the space [s, t] ×  with F = (Fτ )0≤τ ≤t . By Theorem 5.1.1, with f ≡ 0 and g ≡ 1, it follows that sup{E|u|2 : u ∈ decFt Js,t (KF )} = ∞. If Js,t (KF ) would be square integrably bounded, then by Theorem 3.3.5 of Chapter 3, it would be implied that decFt Js,t (KF ) is a bounded subset of the space L2 (, Ft , R). Contradiction.   We shall now consider another consequences of the unboundedness of the set decFt Js,t (KF ) for every fixed 0 ≤ s < t < ∞. First of all, let us note that the unboundedness of decFt Js,t (KF ) implies the unboundedness of the set decFt Js,t (co KF ), because decFt Js,t (co KF ) contains unbounded subset decFt Js,t (KF ). Later on, by the unboundedness of the set decFt Js,t (KF ) it follows that for every α > 0 a set decFt Js,t (α · KF ) is an unbounded subset of L2 (, Ft , R), because by Theorem 3.3.4 of Chapter 3, one has decFt Js,t (α·KF ) = α · decFt Js,t (KF ). Hence in particular, it follows that for every β ∈ (0, 1) a set decFt Js,t (HF ), where HF = dec t {−1, −β} is unbounded. Indeed, let F Fβ (τ, ω) = {β, 1} for (τ, ω) ∈ [0, t] × . Then Fβ (τ, ω) − β = {0, 1 − β} for (τ, ω) ∈ [0, t] × , which by Corollary 2.3.2 of Chapter 2, implies that SF (Fβ − β) = dec t {0, 1 − β} = (1 − β) · KF . Thus the set decFt Js,t (SF (Fβ − F β)) = (1 − β) · decFt Js,t (KF ) is unbounded, because decFt Js,t (KF ) is an unbounded subset of L2 (, Ft , R). Hence it follows that decFt Js,t (SF (Fβ )) is unbounded, because Js,t (SF (Fβ )) = Js,t (SF (Fβ − β)) + Js,t (β), which implies that decFt Js,t (SF [(−1) · Fβ ]) is an unbounded subset of L2 (, Ft , R). But SF [(−1) · Fβ ] = HF . Then the set decFt Js,t (HF ) is an unbounded subset of L2 (, Ft , R). We consider now some further consequences of the unboundedness of the set decFt Js,t (KF ) for fixed 0 ≤ s < t < ∞ with KF = {1D : D ∈ F }. Lemma 5.1.5 Let α > 0, and let h ∈ L2 (R+ × , F , R) be such that hτ (ω) ≥ α for a.e. (τ, ω) ∈ [0, t] × . A set decFt Js,t (h · KF ) is an unbounded subset of the space L2 (, Ft , R) for every fixed 0 ≤ s < t < ∞. Proof By the definition of the set KF we have KF = dec t {0, 1}. Let Fα (τ, ω) = F {0, α}, and Fh (τ, ω) = {0, hτ (ω)} for every (τ, ω) ∈ [0, t] × . We have co Fα (τ, ω) ⊂ co Fh (τ, ω) for every (τ, ω) ∈ [0, t] × . Furthermore, by Corollary 2.3.2 of Chapter 2, one has SF (Fα ) = dec t {0, α} and SF (Fh ) = F

5.1 Itô Set-Valued Functional Integrals

149

dec t {0, h}, which by Theorem 2.3.5 of Chapter 2, implies that co dec t {0, α} = F

F

co SF (Fα ) = SF (co Fα ) ⊂ SF (co Fh ) = co SF (Fh ) = co dec t {0, h}. But, F

co dec t {0, α} = α · co dec t {0, 1} = α · co dec t {0, 1} = α · co KF , and F

F

F

co dec t {0, h} = h · co dec t {0, 1} = h · co dec t {0, 1} = h · co KF . Therefore, F F F α · co KF ⊂ h · co KF . Hence it follows that decFt Js,t (α · co KF ) ⊂ decFt Js,t (h · co KF ), which implies that decFt Js,t (h · co KF ) is an unbounded subset of L2 (, Ft , R), because it contains the unbounded subset decFt Js,t (α · co KF ). Then decFt Js,t (h · KF ) is an unbounded subset of L2 (, Ft , R), because by the boundedness of decFt Js,t (h · KF ) it would be implied that decFt Js,t (h · co KF ) is bounded.   Lemma 5.1.6 Let 0 ≤ s < t < ∞ be fixed, and let h ∈ L2 (R+ × , F , R) be t of the positive measure μ = : dt × P such that there are α > 0, and a set C ∈ F such that 1C h ≥ 1C α, and such that decFt Js,t (1C ·KF ) is an unbounded subset of L2 (, Ft , R). Then decFt Js,t (1C h·KF ) is an unbounded subset of L2 (, Ft , R). Proof Let Gα (τ, ω) = 1C (τ, ω) · {0, α} and Gh (τ, ω) = 1C (τ, ω) · {0, hτ (ω)} for every (τ, ω) ∈ [0, t] × . Similarly as above we get SF (Gα ) = dec t {0, 1C α} F

and SF (Gh ) = dec t {0, 1C h}, which implies that decFt Js,t (1C α · co KF ) ⊂ F decFt Js,t (1C h · co KF ). But, decFt Js,t (1C α · co KF ) = α · decFt Js,t (1C · co KF ) and decFt Js,t (1C · co KF ) is an unbounded subset of L2 (, Ft , R), because by Theorem 3.3.3 of Chapter 3, we have decFt Js,t (1C · co KF ) = co[decFt Js,t (1C · KF )]. Therefore, decFt Js,t (1C α · co KF ) is unbounded, which implies that decFt Js,t (1C h · KF ) is an unbounded subset of L2 (, Ft , R).   Lemma 5.1.7 Let 0 ≤ s < t < ∞ be fixed, and let h ∈ L2 (R+ × , F , R) be t of positive measure μ = : dt × P such that there are β ∈ (0, 1), and a set C ∈ F such that 0 ≤ 1C h < 1C β and such that decFt Js,t (1C · KF ) is an unbounded subset of L2 (, Ft , R). Then decFt Js,t [1C (h − 1) · KF ] is an unbounded subset of the space L2 (, Ft , R). Proof Let Fβ (τ, ω) = 1C (τ, ω) · {−1, −β}, and Fh (τ, ω) = 1C (τ, ω) · {−1, −ht (ω))} for every (τ, ω) ∈ [0, t] × . We have co (Fβ + 1C β)τ (ω) = 1C (τ, ω)co {β − 1, 0} ⊂ 1C (τ, ω)co {hτ (ω) − 1, 0} = co (Fh + 1C h)τ (ω) for every (τ, ω) ∈ [0, t] × , because hτ (ω) − 1 < β − 1 < 0 for every (τ, ω) ∈ C. Therefore, co SF (Fβ + 1C β) = SF (co Fβ + 1C β) ⊂ SF (co Fh + 1C h) = co SF (Fh + 1C h). But, co SF (Fβ + 1C β) = 1C co [dec t {β − F 1, 0}] = 1C (β − 1) co [dec t {1, 0}] = 1C (β − 1) co KF . Similarly, F

co SF (Fh + 1C h) = 1C co [dec t {h − 1, 0}] = 1C (h − 1) co KF . Then 1C (β − F 1) co KF ⊂ 1C (h − 1) co KF , which implies that (β − 1)decF Js,t (1C co KF ) ⊂ decFt Js,t [1C (h − 1) co KF ]. But, (β − 1)decFt Js,t (1C co KF ) is an unbounded subset of L2 (, Ft , R). Therefore, decFt Js,t [1C (h − 1) · co KF ] is unbounded, because it contains the unbounded subset (β − 1)decF Js,t (1C co KF ). Then decFt Js,t [1C (h − 1) · KF ] is an unbounded subset of L2 (, Ft , R).  

150

5 Itô Set-Valued Integrals

t be a set of positive measure Lemma 5.1.8 Let 0 ≤ s < t < ∞ be fixed, C ∈ F 2 + μ = dt × P , and let h ∈ L (R × , F , R) be such that hτ (ω) = 0 for every (τ, ω) ∈ C, and such that decFt Js,t (1C h·co KF ) is unbounded. Then decFt Js,t (h· co KF ) is an unbounded subset of the space L2 (, Ft , R).

Proof Suppose hτ (ω) > 0 for every (τ, ω) ∈ C, and let Fh (τ, ω) = {0, hτ (ω)} and FC (τ, ω) = 1C (τ, ω) · {0, hτ (ω)} for every (τ, ω) ∈ [0, t] × . Then SF (co FC ) ⊂ SF (co Fh ). Hence, similarly as above we obtain co dec t {0, 1C h} = F

co SF (FC ) = SF (co FC ) ⊂ SF (co Fh ) = co SF (Fh ) = co dec t {0, h}, which F

implies that 1C h · co KF ⊂ h · co KF , because co dec t {0, 1C h} = 1C h · F

co dec t {0, 1} = 1C h·co KF and co dec t {0, h} = h·co dec t {0, 1} = h·co KF . F F F Then decFt Js,t (1C h·co KF ) ⊂ decFt Js,t (h·co KF ). Thus, decFt Js,t (h·co KF ) is an unbounded subset of the space L2 (, Ft , R), because it contains an unbounded subset decF Js,t (1C h · co KF ). In a similar way, the case with hτ (ω) < 0 for every (τ, ω) ∈ C can be considered. Let C+ = {(t, ω) ∈ C : ht (ω) > 0} and C− = {(t, ω) ∈ C : ht (ω) < 0}. Similarly to the proof of Lemma 3.3.4 of Chapter 3, one gets decF JT (1C h·co KF ) ⊂ decF JT (1C+ h·co KF )+decF JT (1C− h ·co KF ). Therefore, by the unboundedness of the set decF JT (1C h · co KF ), it follows that at least one set decF JT (1C+ h · co KF ) or decF JT (1C− h · co KF ) is unbounded, which similarly as above, implies that decF JT (h · co KF ) is an unbounded subset of the space L2 (, F, R).   t -measurable Lemma 5.1.9 For every fixed 0 ≤ s < t < ∞, and every finite F partition {C1 , . . . , Cn } of [s, t] ×  there is a set C ∈ {C1 , . . . , Cn } such that decFt Js,t (1C · KF ) is an unbounded subset of the space L2 (, Ft , R).

Proof By Corollary 5.1.1, and Theorem 3.3.5 of Chapter 3, the set  decFt Js,t (KF ) is an unbounded subset of L2 (, F, R). But, decFt Js,t (KF )  ⊂ ni=1 decFt Js,t n n (1Ci · KF i ) · 1D : D ∈ F } = { i ∩D : D ∈ i=1 1C ),n because KF = {( i=1 1C n F } ⊂ i=1 {1Ci ∩D : D ∈ F } = i=1 1Ci · {1D : D ∈ F } = ni=1 1Ci · KF . Hence, by the unboundedness of the set decFt Js,t (KF ), it follows that at least one set of the family {decFt Js,t (1Ci · KF ) : i = 1, . . . , n} has to be unbounded. Therefore, there is a set C ∈ {C1 , . . . , Cn } such that decFt Js,t (1C · KF ) is an unbounded subset of L2 (, Ft , R).  

5.2 Itô Set-Valued Integrals By properties of the Lebesgue set-valued stochastic integrals it follows that for every nonempty decomposable set K ⊂ L2 ([s, t] × , βt ⊗ Ft , Rd ) one has t SFt {() s Kdτ )} = cl[Js,t (K)]. Unfortunately, such result does not hold true for Itô set-valued functional integrals. From (vi) of Lemma 5.1.1, it follows that for fixed 0 ≤ s < t < ∞, and every nonempty decomposable subset K of the t , Rd×m ), containing more than one element, it does not space L2 ([s, t] × , F

5.2 Itô Set-Valued Integrals

151

exist an Ft -measurable set-valued random variable  :  → Cl(Rd×m ) such that cl[J (K)] = SFt (). Then it is impossible to define the Itô set-valued integral  t s,t s KdBτ immediately by the Itô set-valued functional integral Js,t (K). Therefore, we have to apply in such definition, a closed decomposable set decFt Js,t (K) instead of cl[Js,t (K)]. This observation leads to the following definition. Given a nonempty set G ⊂ L2 (R+ × , F , Rd×m ), and an m-dimensional t F-Brownian motion B = (Bt )t≥0 , the Itô set-valued integral s Gdτ of the set G t over the interval [s, t] is defined to be the set-valued random variable s Gdτ : t  → Cl(Rd ) such that SFt ( s Gdτ ) = decFt Js,t (G), where similarly as above t t SFt ( s Gdτ ) denotes the set of all Ft -measurable selectors of s Gdτ . In particular, if G is the closed decomposable subset of the space L2 (R+ × , F , Rd×m ), then by Theorem 2.3.3 of Chapter 2, there exists an F-non-anticipative set-valued process F : R+ ×  → Cl(Rd×m ) such thatG = SF (F ). In such a case, the Itô set-valued t t integral s SF (F )dBτ is denoted by s Fτ dBτ and called the Itô set-valued integral of an F-non-anticipative set-valued process F : R+ ×  → Cl(Rd×m ). From the properties of Itô set-valued functional integrals, the following results follow. ⊂ Theorem 5.2.1 For every fixed 0 ≤ s < t < ∞, and every nonempty sets G, G L2 (R+ × , F , Rd×m ) we have t t (i) s cl(G)dBτ = s G dBτ , a.s., t t (ii) s co(G)dBτ = co s G dBτ a.s., β t (iii) if 0 ≤ s ≤ α < β ≤ t < ∞, then s 1[α,β] (τ ) · G dBτ = α G dBτ , a.s., is square integrably bounded, are convex and Js,t (G) or Js,t (G) (iv) if G and G   t dBτ a.s. τ = t G dBτ + t G then s (G + G)dB s s Proof (i) From (ii) of Lemma 5.1.1, and Lemma 3.3.2 of Chapter 3, we get decFt [Js,t {cl(G)}] = decFt [cl{Js,t (G)] = decFt [Js,t (G)], which by the t definition of Itô set-valued integrals, implies that SFt ( s cl(G)dBτ ) = t SFt ( s G dBτ ). Hence, by Corollary 2.3.1 of Chapter 2, it follows that t t s cl(G)dBτ = s G dBτ a.s. (ii) By virtue of Lemma 5.1.1, the definitionof Itô set-valued integrals and Corolt lary 3.3.2 of Chapter 3, one gets SFt ( s co (G)dBτ ) = decFt Js,t (co G) = t decFt [coJs,t (G)] = co[decFt Js,t (G)] = co SFt ( s G dBτ ). Furthermore, by t t Theorem 2.3.5 of Chapter 2, one has co SFt ( s G dBτ ) = SFt (co s G dBτ ). t t But, immediately from (i) it follows that s co (G)dBτ = s co (G)dBτ a.s. t t Therefore, SFt ( s co (G)dBτ ) = SFt (co s G dBτ ), which by Corollary 2.3.1 t t of Chapter 2, implies that s co (G)dBτ = co s G dBτ a.s. (iii) Let 0 ≤ s ≤ α < β ≤ t < ∞. By virtue of Lemma 5.1.1, one has Js,t (1[α,β] · G) = Jα,β (G). Then decFβ Jα,β (G) = decFβ Js,t (1[α,β] · G). But, decFβ Jα,β (G) ⊂ decFβ [decFt Jα,β (1[α,β] G)]. Therefore, by Corollary 2.3.1

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5 Itô Set-Valued Integrals

of Chapter 2, we get Jα,β (G) ⊂ decFβ Js,t (1[α,β] · G). Hence it follows that decFβ Jα,β (G) ⊂ decFt Js,t (1[α,β] · G), which by the definition of the Itô setβ t valued integral, implies that SFβ ( α G dBτ ) ⊂ SFt ( s 1[α,β] · G dBτ ). Then β t by Corollary 2.3.1 of Chapter 2, it follows that α G dBτ ⊂ s 1[α,β] · G dBτ . On the other hand, by the equality Js,t (1[α,β] · G) = Jα,β (G), it follows that

 SFt

t s

1[α,β] (τ )G dBτ

= decFt Js,t (1[α,β] · G) = decFt Jα,β (G) ⊂

 decFt [decFβ Jα,β (G)] = decFt SFβ

β



 ⊂ SFt

G dBτ

α

β

G dBτ ,

α

t which by Corollary 2.3.1 of Chapter 2, implies that s 1[α,β] (τ )G dBτ ⊂ t β β α G dBτ a.s. Then s 1[α,β] (τ )G dBτ = α G dBτ a.s. are (iv) By Lemma 3.3.3 of Chapter 3, decFt Js,t (G) and decFt Js,t (G) 2 d closed convex subsets of the space L (, Ft , R ), which by Mazur and ˘ Eberlein–Smulian theorems, implies that they are weakly closed. Let 2 m ∈ L (, Ft , R) be such that |u(ω)| ≤ m(ω) for every u ∈ Js,t (G) and a.e. ω ∈ . But, EdecFt Js,t (G)2 = sup{E|z|2 : z ∈ decFt Js,t (G)} ≤ N 2 sup{E[1Ak |Js,t (gk )|2 ] : (Ak )N k=1 ∈ (, Ft ), (gk )k=1 ⊂ G} ≤ Em < ∞. Therefore, by Theorem 3.3.5 of Chapter 3, decFt Js,t (G) is a bounded subset of the space L2 (, Ft , Rd ). Then by Theorem 3.3.4 of Chapter 3, one has = decF Js,t (G) + decF Js,t (G), because decF Js,t (G) is decFt Js,t (G + G) t t t is weakly closed. Therefore, weakly compact and decFt Js,t (G)

 SFt

t s



 t

 t G dBτ (G + G)dBτ = SFt G dBτ + SFt

 = SFt

which implies that

t

s (G

τ = + G)dB

s

t

s

t

G dBτ +

s

t

s



G dBτ +

G dBτ ,

s

t s

dBτ a.s. G

 

Theorem 5.2.2 If (, F, P ) is separable, then for every 0 ≤ s < t < ∞, and every nonempty set G ⊂ L2 (R+ × , F, Rd×m ) there exists a sequence t t n (g n )∞ n=1 ⊂ G such that ( s G dBτ )(ω) = cl{( s gτ dBτ )(ω) : n ≥ 1} for a.e. ω ∈ . Proof By virtue of Lemma 5.1.1, there is a sequence (g n )∞ n=1 ⊂ G such that cl{Js,t (G)} = cl{Js,t (g n ) : n ≥ 1}, which by Lemma 3.3.2 of Chapter 3, implies that decFt (Js,t (G)) = decFt (cl{Js,t (G)}) = decFt ({Js,t (g n ) : n ≥ 1}). Let t (ω) = cl{( s gτn dBτ )(ω) : n ≥ 1} for every ω ∈ . We have  ∈ A(, Rd ), because Js,t (g n ) ∈ SFt () for every n ≥ 1. Then by Corollary 2.3.2 of Chapter 2,

5.2 Itô Set-Valued Integrals

153

we have SFt () = decFt {Js,t (g n ) : n ≥ 1}. On the other hand, by the definition t of the Itô set-valued integral, we have SFt ( s G dBτ ) = decFt [Js,t (G)]. Then t SFt ( s G dBτ ) = SFt (), which by Corollary 2.3.1 of Chapter 2, implies that t t   ( s G dBτ )(ω) = (ω) = cl{( s gτn dBτ )(ω) : n ≥ 1} for a.e. ω ∈ . Theorem 5.2.3 For every 0 ≤ s < t < ∞, and G = {g n : n ≥ 1} ⊂ t t L2 (R+ × , F , Rd×m ), one has s G dBτ = Lim s Gp dBτ a.s., where Gp = 1 p {g , . . . , g } for every p ≥ 1. Proof By virtue of Lemma 5.1.3, we have decFt (Js,t (G)) = Lim decFt [Js,t (Gp )], t t which implies that SFt ( s G dBτ ) = Lim SFt ( s Gp dBτ ). To conclude the proof t t it has to verified that Lim SFt ( s Gp dBτ ) = SFt (Lim s Gp dBτ ). To see that t t let us note that it is clear that Lim SFt ( s Gp dBτ ) ⊂ SFt (Lim s Gp dBτ ). Let t t t a ∈ SFt (Lim s Gp dBτ ), i.e., let a ∈ Lim s Gp dBτ = Li s Gp dBτ a.s. By t Corollary 1.3.4 of Chapter 1, it follows that dist(a, s Gp dBτ ) → 0 a.s. as p → t ∞. But, a sequence {dist(a, s Gp dBτ )}∞ p=1 is integrably bounded by a function t t t 1 ϕ := dist(a, 0 gt dBt ), because dist(a, s Gp+1 dBt ) ≤ dist(a, s Gp dBτ ) for t every p ≥ 1. Therefore, E{dist(a, s Gp dBτ )} → 0 as p → ∞. By Theorem 2.3.4 of Chapter 2, we have

 t = dist2 a, SFt Gp dBτ s



 inf | a − u | : u ∈ SFt 2

s

t

 Gp dBτ





= E dist

2



t

a,

 Gp dBτ

s

for every p ≥ 1, where | · | is the norm of the space L2 (, Ft , R d ). Then t t dist(a, SFt ( s Gp dBτ )) → 0 as p → ∞, for every a ∈ SFt (Lim s Gp Bτ ). t t Therefore, for every a ∈ SFt (Lim s Gp Bτ ), one has a ∈ Li SFt ( s Gp dBτ ) = t t t Lim SFt ( s Gp dBτ ). Thus SFt (Lim s Gp dBτ ) ⊂ Lim SFt ( s Gp dBτ ), which t t implies that SFt (Lim s Gp dBτ ) = Lim SFt ( s Gp dBτ ). This and the t t t equality SFt ( s G dBτ ) = Lim[SFt ( s Gp dBτ )] imply that SFt ( s G dBτ ) = t SFt (Lim s G p dBτ ), which by Corollary 2.3.1 of Chapter 2, ends the proof.   Corollary 5.2.1 If (, F, P ) is a separable probability space, and G is a 2 + d×m ), then there exists nonempty closed subset of the space L  t (R × , F , Rt ∞ n a sequence (g )n=1 ⊂ G such that s G dBτ = Lim s Gp dBτ a.s. for every 0 ≤ s 0, x ∈ Rd , and u ∈ UT there exists the unique strong solution zux to (8.1.1) defined on [0, T ] × , satisfying the initial condition zux (0) = x a.s. Then we can define on UT an operator λx by setting λx (u) = zux . We have λx (u) ∈ CF (T , Rd ) for every x ∈ Rd , and u ∈ UT , where (CF (T , Rd ), | · | ) is the Banach space defined in Chapter 6. Lemma 8.1.1 Let PF = (, F, F, P ) be a complete filtered probability space with a filtration F = (Ft )t≥0 satisfying the usual conditions and let B = (Bt )t≥0 be an m-dimensional F-Brownian motion on PF . If f and g are measurable functions satisfying conditions (H), then λx is a continuous mapping on UT depending continuously on x ∈ Rd . Proof Let u ∈ UT and (un )∞ n=1 be a sequence of UT such that un − u → 0 as n → ∞. By the definition of the mapping λx we have λx (u) = zux and λx (un ) = zux n for n = 1, 2, . . . . Similarly to the proof of Theorem 6.2.2 of Chapter 6, for every 0 ≤ t ≤ T and ε > 0 we get E[ sup |zux n (X) − zux (X)|2 ] ≤ 0≤s≤t



t

ε + 2(T + 4) 0

k(τ )E[ sup |zux n (X) − zux (X)|2 ]dτ, 0≤s≤τ

which by Gronwall’s lemma implies that E[sup0≤s≤t |zux n (X) − zux (X)|2 ] ≤ t ε exp(2(T +4) 0 k(τ )dτ ) for every n ≥ 1. Therefore, limn→∞ E[sup0≤s≤t |zux n (s)− zux (s)|2 ] ≤ ε for every ε > 0 and 0 ≤ t ≤ T . Thus, limn→∞ E[sup0≤s≤T |zux n (X) − zux (X)|2 ] = 0. Then | λx (un ) − λx (u) | → 0 as n → ∞. In a similar way for every x ∈ Rd , every sequence (xn )∞ n=1 convergent to x, and a fixed u ∈ UT , we get | λxn (u) − λx (u) | → 0 as n → ∞.   We can prove now the following existence theorem. Theorem 8.1.1 Let f and g be measurable functions and satisfy conditions (H). If K : R+ × Rd → R and  : R+ × Rd × U → R are continuous, then for every bounded domain D, a filtered probability space PF , an m-dimensional FBrownian motion B = (Bt )t≥0 defined on PF , and x ∈ D, there exists u¯ ∈ UT such that JD (u, ¯ zux¯ ) = sup{JD (u, zux ) : u ∈ UT }. Proof Let us note that by virtue of Lemma 8.1.1 and Theorem 1.7.1 of Chapter 1, a functional UT  u → JD (u, λx (u)) ∈ R is continuous. Then α = : sup{JD (u, λx (u)) : u ∈ UT } < ∞. By the definition of λx we have α = sup{JD (u, zux ) : u ∈ UT } < ∞. Then there exists a sequence (un )∞ n=1 ⊂ UT such

252

8 Stochastic Optimal Control Problems

that α = limn→∞ JD (un , λx (un )). By compactness of UT there exists subsequence ∞ (unk )∞ ¯ ∈ UT such that unk − u ¯ T → 0 as k → ∞ . By virtue k=1 of (un )n=1 , and u of Lemma 8.1.1 it follows that | λx (unk ) − λx (u) ¯ | → 0 as k → ∞ . Furthermore, by the definitions of λx , and the definition of the norm | · | it follows that there x ∞ exists a subsequence, for simplicity denoted by (zkx )∞ k=1 , of the sequence (zn )k=1 x x x such that sup0≤t≤T |zk − zu¯ | → 0 a.s. as k → ∞ , where zu¯ = λx (u). ¯ By virtue of Theorem 1.7.1 of Chapter 1, we have τDnk → τ¯D a.s. as k → ∞ , where τDnk and τ¯D denote the first exit times of zux n and zux¯ , respectively, from a domain D. k Let us note that in a particular case, if znxk (t) ∈ D for every 0 ≤ t ≤ T , then we have τDnk = T a.s. By continuity of functions , and K, and Theorem 1.7.2 of Chapter 1, it follows that α = limk→∞ JD (unk , λx (unk )) = JD (u, ¯ λx (u)), ¯ i.e., that α = JD (u, ¯ zux¯ ). Then (u, ¯ zux¯ ) is an optimal pair for (8.1.3).   We can consider the above optimal control problem with some special type of controls v x = (vtx )t≥0 of the form vitx = ϕ(t, zux (t)) for every u ∈ UT a.s. for t ≥ 0, where ϕ : R+ × Rd → U ⊂ Rk is a given measurable function. Such type controls are called Markov’s control, because with such v the corresponding process zvx = (zvx (t))t≥0 becomes an Itô diffusion, and in particular a Markov process. From the proof of Theorem 8.1.1 it follows that existence of the optimal pair to the optimal control problem (8.1.3) depends on the compactness of the set UT . But, the optimal control problem (8.1.3) can be also considered with a set VT of CT -random variables with CT = C([0, T ], Rk ). It is difficult to select such sets to be compact in the norm topology of the space L(, F, CT ). Therefore, application of Theorem 8.1.1, to optimal control problems of the form (8.1.3) with VT ⊂ L2 (, F, CT ) is not possible. But, sets VT ⊂ L2 (, F, CT ) can be selected to be weakly compact in distribution. Therefore, we can consider the optimal control problem (8.1.3) with control parameters belonging to weakly compact in distribution sets of CT -random variables u such that ut ∈ U a.s for every t ∈ [0, T ]. Let VT be a weakly compact in distribution set of CT -random variables such that ut (ω) ∈ U for every u ∈ VT , every 0 ≤ t ≤ T , and a.e. ω ∈ u , where (u , Fu , Pu ) is a probability space such that u : u → CT is (Fu , β(CT ))measurable. We can define a filtration Fu = (Ftu )0≤t≤T on (u , Fu , Pu ) by setting Ftu = σ ({us : 0 ≤ s ≤ t}) for every 0 ≤ t ≤ T . Unfortunately, in the general case a filtered probability space PFu u = (u , Fu , Fu , Pu ) may not be rich enough to support an Fu -Brownian motion B u = (Btu )0≤t≤T . Therefore, we must extend (u , Fu , Fu , Pu ) (see [48], p.42) in order to construct this. For simplicity, we shall assume that VT is a weakly compact in distribution set of CT -random variables such that ut ∈ U a.s. for every 0 ≤ t ≤ T , and such that for every u ∈ VT there exists an m-dimensional Fu -Brownian motion B u = (Btu )0≤t≤T on (u , Fu , Fu , Pu ). In such a case, for a fixed x ∈ D and measurable functions f and g satisfying conditions (H), and every u ∈ VT there exists (see [83], Th.11.2.1 of Chap. 11) exactly one strong solution zux to the stochastic differential equation 

t

zt = x + 0



t

f (τ, uτ , xτ )dτ + 0

g(τ, uτ , xτ )dBτu

(8.1.4)

8.1 Optimal Control Problems for Systems Described by SDE(f, g)

253

x (V ) = {(P u , B u , zx ) : u ∈ V}, and Z x (V ) = such that P z0−1 = μx . Let WD T u D T Fu x {X : (PF , B, X) ∈ WD (VT )}, where PFu u = (u , Fu , Fu , Pu ) for every u ∈ VT . Similarly to the proof of Theorem 6.3.1 of Chapter 6, the following result can be obtained.

Lemma 8.1.2 Let D be a nonempty bounded domain of the space Rd , and let U be a nonempty closed subset of Rk . If f and g are measurable and satisfy conditions (H), and VT is a weakly compact in distribution set of CT -random variables such that for every for every u ∈ VT one has ut (ω) ∈ U for every 0 ≤ t ≤ T and a.e. x (V ) is nonempty and weakly compact in ω ∈ u , then for every x ∈ D the set ZD T distribution. x (U ) = ∅, because by [83, Th.11.2.1 of Chap. 11], for Proof It is clear that ZD T every u ∈ VT there exists a weak solution (PFu u , B u , zux ) to (8.1.4) satisfying the initial condition zux (0) = x, Pu -a.s. Similarly to the proof of Theorem 6.3.1 x (V ) is weakly compact in of Chapter 6, it can be verified that the set ZD T ∞ distribution. Indeed, for every sequence (un )n=1 of VT there exists a sequence x x (zux n )∞ n=1 of ZD (VT ). By the definition of the set ZD (VT ) there exists a sequence un n n x (B n )∞ n=1 of m-dimensional Fun -Brownian motions B such that (PFun , B , zun ) ∈ x WD (VT ). Similarly to the proof of Theorem 6.3.1 of Chapter 6, we can verify ), an increasing sequence , F, that there exists a filtered probability space ( F, P k+m+d )of positive integers (nk )k=1 , a sequence (u¯ n , B¯ n , zux¯ n )∞ n=1 of C([0, T ], R random variables and an C([0, T ], Rk+m+d )-random variable (u, B, z) defined on d , P ), such that (unk , B nk , zux ) = , F (u¯ nk , B¯ nk , zux¯ n ) for every k ≥ 1, and such ( nk k that sup0≤t≤T |u¯ nk (t) − u(t)| + sup0≤t≤T |B¯ tnk − Bt | + sup0≤t≤T |zx (t) − zt | → 0 u¯ nk

-a.s. as k → ∞. It is clear that u(t) ∈ U , P -a.s. for every 0 ≤ t ≤ T . Hence, P -a.s., similarly to the proof of Theorem 6.3.1 of Chapter 6, it follows that z0 = x, P and such that  t  t f (τ, uτ , zτ )dτ + g(τ, uτ , zτ )dBτ zt = z 0 + 0

0

-a.s. and every 0 ≤ t ≤ T . Then z ∈ Z x (UT ). P D

 

We can now prove the following existence theorem. Theorem 8.1.2 Let D and U be such as in Lemma 8.1.2, and assume that f and g are measurable and satisfy conditions (H). Let VT be a weakly compact in distribution set of CT -random variables such that for every u ∈ VT , one has ut (ω) ∈ U for every 0 ≤ t ≤ T and a.e. ω ∈ u . If K : R+ × Rd → R and  : R+ × Rd × U → R are continuous and bounded, then for every x ∈ D there ¯ F, ¯ P¯ ), an m-dimensional F-Brownian ¯ ¯ F, exists a filtered probability space (, ¯ an CT -random variable u, motion B, ¯ and a strong solution zux¯ to (8.1.4) defined on ¯ such that J D (u, ¯ F¯ , P¯ ) corresponding to (u, ¯ zux¯ ) = sup{JD (u, X) : (u, X) ∈ (, ¯ B) x (V )}. VT × ZD T

254

8 Stochastic Optimal Control Problems

x (V ) is nonempty weakly Proof By virtue of Lemma 8.1.2, it follows that ZD T compact in distribution. By the boundedness of functions  and K it follows that x (V )} < ∞. Then there exists a sequence α = sup{JDx (u, zux ) : (u, zux ) ∈ VT × ZD T x ∞ x {(un , zun )}n=1 of the set VT × ZD (VT ) such that α = : limn→∞ JD (un , zux n ). It x (V ) is weakly compact in distribution. Similarly to the is clear that VT × ZD T proof of Lemma 8.1.2, we can verify that there exists a filtered probability space ¯ P¯ ), a sequence {(u¯ n , B¯ n , zx )}∞ of C([0, T ], Rk+m+d )-random vari¯ F¯ , F, (, u¯ n n=1 ¯ P¯ ) ¯ zx ) defined on (, ¯ F, ¯ B, ables, and an C([0, T ], Rk+m+d )-random variable (u, u¯ d such that (un , B n , zux n ) = (u¯ n , B¯ n , zux¯ n ) for every n ≥ 1, one has sup0≤t≤T |u¯ n (t) − u(t)| ¯ + sup0≤t≤T |B¯ tn − B¯ t | + sup0≤t≤T |zux¯ n (t) − zux¯ (t)| → 0 P¯ -a.s. as n → ∞, zx (0) = x, P¯ -a.s., and such that u¯

 zux¯ (t) = zux¯ (0) +

t 0

 f (τ, u¯ τ , zux¯ (τ ))dτ +

t 0

g(τ, u¯ τ , zux¯ (τ ))d B¯ τ ,

P¯ -a.s. and every 0 ≤ t ≤ T . Furthermore, for every n ≥ 1 one has |α−J D (u, ¯ zux¯ )| ≤ x x x x |α − J D (u¯ n , zu¯ n )| + |J D (u¯ n , zu¯ n ) − J D (u, ¯ zu¯ )|, where J D (u, ¯ zu¯ ) is defined by J D (u, z) = E



τD

 (t, z(t), u(t))dt + K( τD , z( τD ))

0

u, z) defined on a probability space for C([0, T ], Rk+d )-random variable ( d ¯ ¯ ¯ (, F , P ), where τD is the first exit time of z from the set D. But, (un , zux n ) = (u¯ n , zux¯ n ) for every n ≥ 1. Therefore, by continuity of functions K and , one gets |α − J D (u¯ n , zux¯ n )| = |α − JD (un , zux n )|. But α = limn→∞ JD (un , zux n ). Then limn→∞ |α − J D (u¯ n , zux¯ n )| = 0. Furthermore, for every n ≥ 1 we have |J D (u¯ n , zux¯ n ) − J D (u, ¯ zux¯ )| ≤ E¯   ¯ E 

n τ¯D

0

 0

T

|(t, zux¯ n (t), u¯ n (t) − (t, zux¯ (t), u(t))|dt+ ¯ 

(t, zux¯ (t), u(t))dt ¯

− 0

τ¯D

 

+ (t, zux¯ (t), u(t))dt ¯ 

¯ ¯ E|K( τ¯Dn , zux¯ n (τ¯Dn )) − K(τ¯Dn , zux¯ (τ¯Dn ))| + E|K( τ¯Dn , zux¯ (τ¯Dn )) − K(τ¯D , zux¯ (τ¯D ))|, zx

zx

where τ¯Dn = τDu¯ n and τ¯D = τDu¯ . Therefore, by virtue of Theorems 1.7.1 and 1.7.2 of Chapter 1, it follows that α = J D (u, ¯ zux¯ ) = sup{JDx (u, zux ) : (u, zux ) ∈ VT × x ZD (VT )}. Thus, u¯ is an optimal control to the optimal control problem (8.1.3).  

8.2 Optimal Control Problems for Systems Described by SDI (F, G )

255

8.2 Optimal Control Problems for Systems Described by SDI (F, G ) We shall now extend the above optimal control problem (8.1.3) on the case in which the dynamics of a control system is described by the initial value problem of the form t t  Xt − Xs ∈ cl{ s F (τ, Xτ )dτ + s co (G ◦ X)dBτ } (8.2.1) P X0−1 = μx a.s. Similarly as above it has to be satisfied a.s. for every 0 ≤ s < t < ∞ by its weak solution (PF , B, X), with the probability measure μx : β(Rd ) → [0, 1] defined in Remark 6.3.1 of Chapter 6. The performance functional, depending only on the weak solution (PF , X, B) to the initial value problem (8.2.1), is defined by ' JDx (X)

=E s

X τD

( (t, Xt )dt

+ K(τDX , Xτ X ) D

,

: R+ × Rd → R and with x ∈ D, where D is a bounded domain of Rd and + d K : R × R → R are given continuous bounded functions. Such optimal control problem is denoted by ⎧ t t ⎪ ⎨ Xt − Xs ∈ cl{ s F (τ, Xτ )dτ + s co (G ◦ X)dBτ } X0 = x a.s. x (F,G ) ⎪ ZD ⎩ x JD (X) −→ max

(8.2.2)

and called an optimal control problem for the control system described by the stochastic functional inclusion SDI (F, G) . By a solution to the optimal control , X, B) to (8.2.1) such that J x (X) = problem (8.2.2) we mean a weak solution (P F D x x x sup{JD (X) : X ∈ ZD (F, G)}, where ZD (F, G) = {X : (PF , B, X) ∈ x (F, G) is said to be an admissible set to W(F, G, μx ). In such case the set ZD X) the optimal control problem (8.2.2). If there exists a weak solution (PF , B, x x x to (8.2.1) such that X ∈ ZD (F, G), and such that JD (X) = sup{JD (X) : X ∈ x (F, G)}, then (P , B, is called an optimal solution to the optimal control ZD X) F problem (8.2.2). We can prove the following existence theorem. Theorem 8.2.1 Let F : R+ × Rd → Cl(Rd ) be a continuous convex-valued bounded multifunction, and let G = {g n : n ≥ 1} be a family of continuous functions g n : R+ × Rd → Rd×m such that there exists a sequence (cn )∞ n=1 of ∞ 2 n positive numbers with n=1 cn < ∞, and such that |g (t, x)| ≤ cn for every n ≥ 1 and (t, x) ∈ R+ × Rd . If K : R+ × Rd → R and  : R+ × Rd → R are continuous and bounded functions, then for every nonempty bounded domain

256

8 Stochastic Optimal Control Problems

D ⊂ Rd and every x ∈ D the optimal control problem (8.2.2) possesses an optimal solution. Proof Similarly to the proof of Theorem 6.3.1 of Chapter 6, we can verify that the x (F, G) is weakly compact in distribution. By the boundedness of functions set ZD x (F, G)} < ∞. Therefore, K and  it follows that α = : sup{JDx (X) : X ∈ ZD x ∞ there exists a sequence (Xn )n=1 of ZD (F, G) such that α = limn→∞ JDx (Xn ). By x (F, G), there exist sequences (P n )∞ and (B n )∞ the definition of the set ZD n=1 Fn n=1 of filtered probability spaces PFn n = (n , Fn , Fn , Pn ) and m-dimensional Fn Brownian motions B n = (Btn )t≥0 on PFn n , such that (PFn n , B n , Xn ) ∈ W(F, G, μx ). Similarly to the proof of Theorem 6.3.1 of Chapter 6, it follows that there exists , ), an m-dimensional , F a filtered probability space ( F, P F-Brownian motion d sequences (B n )∞ and (X n )∞ of B = (Bt )t≥0 , an C(R )-random variable X, n=1 n=1 d ) such that , F, P F-Brownian motions, and C(R )-random variables defined on ( d n n , B)|+ρ n , X) → 0, P -a.s. as n → ∞. Hence, , Xn ) and ρm (B (B n , Xn ) = (B d (X similarly to the proof of Theorem 6.3.1 of Chapter 6, it follows that   ⎧   ⎨X s ∈ cl t F (τ, X τ )dτ + t co (G ◦ X)d B τ t − X s s ⎩ X0 = x a.s. -a.s. for every 0 ≤ s < t < ∞. Similarly to the proof of Theorem 8.1.2, is satisfied P , B, Then there exists an optimal weak solution (P X) we get α = JDx (X). F = sup{J x (X) : X ∈ Z x (F, G)}. to (8.2.1) such that JDx (X)   D D It is natural to ask if immediately from Theorem 8.2.1, the existence of an optimal pair (u, zux ) to the optimal control problem (8.1.3) follows. Unfortunately, in the general case the answer to such question in negative, because to get such result we have to consider the stochastic differential inclusion of the form t t  Xt − Xs ∈ cl{ s F (τ, Xτ )dτ + s SF (G ◦ X)Bτ } (8.2.3) P X0−1 = μx a.s. with F (t, x) = {f (t, x, u) : u ∈ U } and G(t, x) = {g(t, x, u) : u ∈ U } for every (t, x) ∈ R+ × Rd . But SF (G ◦ X) is a decomposable subset of the space d×m L2 (R+ × , F , R  t ). Then by Corollary 5.3.2 of Chapter 5, the set-valued stochastic integral s SF (G ◦ X)Bτ is square integrably bounded if and only if SF (G ◦ X) is a singleton. Therefore, we can only consider the stochastic optimal control problem (8.2.2) with the stochastic differential inclusion of the form 

t t Xt − Xs ∈ s F (τ, Xτ )dτ + s g(τ, Xτ )Bτ P X0−1 = μx a.s.

(8.2.4)

8.3 Notes and Remarks

257

On the other hand, by Theorem 4.5.2 of Chapter 4, for a given continuous stochastic process X = (Xt )t≥0 , satisfying the above inclusion a.s. for every 0 ≤ s < t < t ∞, there is a measurable selector f of F ◦ X such that Xt − Xs = s fτ dτ + t g(τ, Xτ )Bτ a.s. for every 0 ≤ s < t < ∞. It leads to a contradiction, because st s fτ dτ is measurable and has not to be F-adapted. Then we can only look for continuous solutions to the above inclusion. In such a case the stochastic process g  t◦ X will be measurable but not necessary an F-non-anticipative, and the integral Therefore, we can only consider system (8.2.4) s g(τ, Xτ )dBτ maybe not defined. t t with the stochastic integral s g(τ, ·)dBτ instead of s g(τ, Xτ )dBτ . In such case we can look for a continuous solution X that does not need to be F-adapted. For such + defined solution X, by Theorem 2.2.4 of Chapter 2 there is a process t  t u : R × → k R such that ut (ω) ∈ U and Xt − Xs = s f (τ, Xτ , uτ )dτ + s g(τ, ·)Bτ . The above problem possesses the positive solution (see [48], Th.22 of Chap. 7) if a dynamic system is described by stochastic functional inclusions of the form Xt − Xs ∈ cl{Js,t (SF (F ◦ x)) + Js,t (SF (G ◦ x))}

(8.2.5)

with multifunctions F and G defined such as above. But, the optimal control problem with the dynamic system described by the above stochastic functional inclusion possesses an optimal solution if among others a function (g · g ∗ )(t, x, ·) is affine. Such requirement is too strong for practical applications, because a function g such that g(t, x, ·) is linear, does not imply that (g · g ∗ )(t, x, ·) is affine.

8.3 Notes and Remarks Results of this chapter are consequences of compactness and weak compactness in distributions of sets of control parameters. In a similar way as above it is possible to consider the above optimal control problems with compact and weak compact control subsets of much general Banach spaces. Lemma 1.1 and Theorem 1.1 have been considered in the monograph [48]. Let us note that the existence of optimal solutions to optimal control problems for dynamical systems described by stochastic functional inclusions of the form (8.2.5) with set-valued mappings F and G defined by F (t, x) = {f (t, x, u) : u ∈ U } and G(t, x) = {g(t, x, u) : u ∈ U } implies the existence of optimal pair (X, u) of F-non-anticipative stochastic processes t t such that Xt − Xs = s f (τ, Xτ , uτ )dτ + s g(τ, Xτ , uτ )dBτ and ut ∈ U a.s. for every 0 ≤ s < t ≤ τDX . It follows immediately from [48, Th.2.1 of Chap. 7, Th.1.5 of Chap. 3], and Theorem 2.2.4 of Chapter 2. Unfortunately, solutions of optimal control problems for dynamical systems described by stochastic differential inclusions SDI (F, G) do not possess of the above property. It can be considered with a family G = {g n : n ≥ 1} consisting of F-non-anticipative stochastic processes do not depend on the pair (x, u) ∈ Rd × Rk and ∞ that n |2 is integrable. In such a case we can look for weak are such that |g n=1 t

258

8 Stochastic Optimal Control Problems

solutions to SDI (F, G) with continuous measurable processes X = (Xt )t≥0 that maybe not to be F-adapted. Why having the result presented by [48, Th.2.1 of Chap. 7] we consider the optimal control problem described by SDI (F, G) that does not generalize the optimal control problem described by stochastic differential equations depending on stochastic parameters? It is therefore, because the methods presented in [48, Th.2.1 of Chap. 7] depends on the convexity of the multifunction G · G∗ , and in the general case this requirement is only satisfied in some special cases. Therefore, methods presented in [48, Th.2.1 of Chap. 7], cannot be applied to a lot of important stochastic optimal control problems. There are a lot of papers (see for example [1]) devoted to optimal control of stochastic dynamical systems. Some problems considered in these papers can be investigated by methods presented in the present chapter.

Chapter 9

Mathematical Finance Problems

Some optimal control problems of Financial Mathematics are presented. In particular, selected problems of optimal pricing and optimal portfolios in a given financial market are considered.

9.1 Market, Portfolio, and Arbitrage The basic notions of Financial Mathematics can be described by some stochastic processes defined on a given filtered probability space PF = (, F, F, P ) with a filtration F = (Ft )0≤t≤T satisfying the usual conditions, where  is a set of all random events, characterized by the filtration F on the time interval [0, T ], that influence on the financial market. From mathematical point of view, by a financial market we mean an (d + 1)-dimensional F-adapted stochastic process X = (Xt )0≤t≤T , defined on a filtered probability space PF such that Xt = (Xt0 , Xt1 , . . . , Xtd ) with  Xt0 (ω) = 1 +

0

t

ρt (ω)Xτ0 (ω)dτ

(9.1.1)

and  Xti (ω) = X0 (ω) +

0

t

μiτ (ω)dτ +

m   j =1 0

t

i,j

σt (ω)dBτj

(9.1.2)

for every i = 1, 2, . . . , d, j = 1, 2, . . . , d, 0 ≤ t ≤ T , and a.e. ω ∈ , where ρ = (ρt )0≤t≤T is a real-valued measurable bounded process, B = (B 1 , . . . , B m )∗ is an F-Brownian motion on PF , and μ = (μ1 , . . . , μd )∗ and σ = [σ ij ]d×m are vector© Springer Nature Switzerland AG 2020 M. Kisielewicz, Set-Valued Stochastic Integrals and Applications, Springer Optimization and Its Applications 157, https://doi.org/10.1007/978-3-030-40329-4_9

259

260

9 Mathematical Finance Problems

and matrix-valued stochastic processes. The first of the above equations defines the asset of the financial market, called the bank account, by its price Xt0 (ω), defined by the process ρ = (ρt )0≤t≤T , called the interest rate. The remaining d assets, called the bond, are “risky.” In what follows, the process X = (X1 , . . . , Xd ) will be defined by the vector stochastic differential equations 

t

Xt (ω) = X0 (ω) +



t

μτ (ω)dτ +

(9.1.3)

σt (ω)dBτ

0

0

for a.e. ω ∈  and 0 ≤ t ≤ T . The prices in the financial market are modeled by the above system (9.1.1) and (9.1.2) of d + 1 linear stochastic differential equations, or by the vector stochastic differential equation (9.1.3), where the vector-valued process μ is called the vector of mean rates of return and the F-adapted matrixvalued process σ is called the dispersion matrix process. All the above processes are assumed to be measurable, F-adapted, and bounded uniformly in (t, ω) ∈ [0, T ] × . It is assumed that all assets (or securities) on the financial marked X = (Xt )0≤t≤T are traded continuously. An (d + 1)-dimensional measurable Fadapted vector-valued process λ = (λt )0≤t≤T , whose components λ0 , λ1 , . . . , λd represent numbers of units of the securities, is called the portfolio in the financial market X = (Xt )0≤t≤T . It is clear that the value Vt (λ) at time t ∈ [0, T ] of the  portfolio λ = (λt )0≤t≤T is defined by Vt (λ) = di=0 λit Xti . Portfolios λ = (λt )0≤t≤T on the financial market X = (Xt )0≤t≤T such that  0

T

 |λ0τ ρτ Xτ0 | +

d 

λiτ μiτ | +

i=1

 d m   [λiτ ]2 [στij ]2 dτ < ∞ a.s. j =1

i=1

and  Vt (λ) = V0 (λ) +

t

λτ dXτ 0

for every t ∈ [0, T ], are called self-financing. A portfolio λ = (λt )0≤t≤T such that λ0t

= V (0) +

d   i=1

t 0

λiτ dXτi

−λ

i

(t)Xti

and which is self-financing is called admissible if the corresponding value process V (λ) = (Vt (λ))0≤t≤T is for a.e. (t, ω) ∈ [0, T ] ×  lower bounded, i.e., if there exists K = : K(λ) < ∞ such that Vt (λ)(ω) ≥ −K for a.e. (t, ω) ∈ [0, T ] × . Finally, an admissible portfolio λ = (λt )0≤t≤T is said to be an arbitrage in the financial market X = (Xt )0≤t≤T if the corresponding value process V (λ) = (Vt (λ))0≤t≤T satisfies V0 (λ) = 0, VT (λ) ≥ 0 a.s. and P ({VT (λ) > 0}) > 0. In other words, a portfolio λ = (λt )0≤t≤T is an arbitrage in the financial market

9.2 Option Pricing and Consumption Processes

261

X = (Xt )0≤t≤T if it gives an increase in the value from time t = 0 to time t = T a.s., and a strictly positive increase with positive probability, i.e., it generates a profit without any risk of losing money. Intuitively, the existence of an arbitrage is a sign of a lack of an equilibrium in the financial market.

9.2 Option Pricing and Consumption Processes We can imagine now an investor who starts with some initial endowment x ≥ 0 and invests it in the portfolio λ = (λt )0≤t≤T with λt = (λ0t , λ1t , . . . , λdt ), where λit denotes the number of shares of assets i owned by the investor at a time t ∈ [0, T ]. If, on the other hand, the investor chooses at time t + h to consume an amount Ct+h , and reduce the wealth accordingly, then its wealth at time t ∈ [0, T ] can be defined by  Vt (λ) = x +

t

[ρ(t)Vτ (λ) − Cτ ]dτ +

0

d   0

i=1 d  d  

t

λiτ στij dBτj

i=1 j =1 0

t

[μiτ − ρτ ]λiτ dτ +

(9.2.1)

a.s.

T A real-valued measurable F-adapted process C = (Ct )0≤t≤T , such that 0 Ct dt < ∞ a.s., is said to be a consumption process. Usually it is assumed that a portfolio  T process λ = (λt )0≤t≤T satisfies di=1 0 [λit ]2 dt < ∞ a.s. A pair (λ, C) of a portfolio process λ, and a consumption process C, is said to be admissible for the initial endowment x ≥ 0 if the wealth process V = (Vt )0≤t≤T defined by (9.2.1) satisfies Vt ≥ 0 a.s. for every t ∈ [0, T ]. T  T Remark 9.2.1 Conditions di=1 0 [λit ]2 dt < ∞ a.s. and 0 Ct dt < ∞ a.s. imply (see [35], Prob.6.15 of Chap. 5) that the stochastic differential equation (9.2.1) has the unique strong solution such that

 Vt (λ) = exp

t



  t ρs ds x + exp −

0

0

 0

t

0

 exp − 0

τ

τ



ρs ds [λ∗τ (μτ − ρτ 1) − Cτ ]dτ +

  ρs ds λ∗τ στ dBτ

for 0 ≤ t ≤ T , where 1 denotes the d-dimensional vector with every components equal to 1. Suppose that at time t = 0 we sign a contract which gives us the option to buy at a specified time T > 0, called the maturity date, one share of stock k at a specified price q, called the exercise price. At maturity date, if the price XTk of stock

262

9 Mathematical Finance Problems

k is below the exercise price, the contract is worthless to us. On the other hand, if XTk > q, we can exercise our option, i.e., we can buy one share at the preassigned price q, and then sell the share immediately in the market for XTk > q. This contract, which is called an option, is equivalent to a payment (μkT − q)+ dollars at maturity. Sometimes the term European option is used to described this financial instrument, in contrast to an American option, is thus equivalent to payment of (XTk − q)+ dollars at maturity. The above concept of option can be generalized as a contingent claim. More precisely, the contingent claim is a financial instrument consisting of : (1) a payoff rate g = (gt )0≤t≤T and (2) a terminal payoff rate fT , where g is a non-negative, measurable F-adapted process, fT is a non-negative FT -measurable T random variable, and for some α > 1 one has E[fT + 0 gt dt]α < ∞. The following contingent claim valuation problem is important from the practical point of view: What is a fair price to pay at time t = 0 for a contingent claim? The answer on this question is connected with the concept of hedging strategy. It is defined as follows. Let x ≥ 0 be given, and let (λ, C) be a portfolio/consumption pair which is admissible for the initial endowment x ≥ 0. The pair (λ, C) is called a hedging strategy against the contingent claim (g, fT ) provided : (1) Ct = gt for 0 ≤ t ≤ T , and (2) VT = fT holds a.s., where V = (Vt )0≤t≤T is the wealth process associated with the pair (λ, C) and with the initial condition V0 = x. The fair price for a contingent claim is the smallest number x ≥ 0 which allows the construction of the hedging strategy with initial wealth x.

9.3 Finance Optimal Control Problems We shall consider now the optimal control problem described by a stochastic differential equation 

dVt = f (t, Vt , ut )dt + g(t, Vt , ut )dBt a.s. for t ∈ [0, T ] , V0 = x a.s.

(9.3.1)

considered together with the performance functional of the form

 JDx (u, V ) = E

0

τD

 (t, Vt , u)dt + K(τD , VτD ) ,

depending on a control process u = (ut )0≤t≤T , with ut = (πt , ϕ, ct ) for every 0 ≤ t ≤ T , where π = (π 0 , π 1 , . . . , π d ) is a portfolio defined above, φ = (φt )0≤t≤T denotes the borrowed process, i.e., φt represents the amount borrowed at t ∈ [0, T ], and c = (ct )0≤t≤T is a consumption process. As usual τD denotes the first exit time of the process V from a bounded domain D ⊂ R. If the functions f : R+ × R+ × U → R and g : R+ × R+ × U → Rm satisfy conditions (H) of Chapter 8, then immediately from Theorem 8.1.1 of Chapter 8 we obtain the following result.

9.3 Finance Optimal Control Problems

263

Theorem 9.3.1 Let D be a nonempty bounded domain of the real line R, and U a nonempty closed subset of Rd+3 . Assume f and g are measurable and satisfy conditions (H) of Chapter 8. If K : R+ × Rd → R and  : R+ × Rd × U → R are continuous bounded, then for every filtered probability space PF , an mdimensional F-Brownian motion B = (Bt )t≥0 defined on PF , and x ∈ D there = (φ t )0≤t≤T , and exists a portfolio π = ( π 0, π 1, . . . , π d ), a borrowed process φ x u, V ) = sup{JDx (u, V ) : u ∈ a consumption process c = ( ct )0≤t≤T such that JD ( UT }, where ut = ( πt , ϕ , ct ) and ut = (πt , ϕ, ct ) for every 0 ≤ t ≤ T and UT ⊂ and V are strong C([0, T ], Rd+3 ) is a compact set defined in Chapter 8, and V solutions to the initial value problem (9.3.1) corresponding to u ∈ UT and u ∈ UT , respectively.   Similarly as in Chapter 8, the above optimal control problem can be considered with control parameters ut = (πt , ϕ, ct ) belonging to a weakly compact in distribution set VT of C([0, T ], Rd+3 )-random variables such that for every u ∈ VT one has ut (ω) ∈ U for every 0 ≤ t ≤ T and a.e. ω ∈ u , where (u , Fu , Pu ) is a probability space such that u : u → C([0, T ], Rd+3 ) is (Fu , βC )-measurable. Similarly as above a filtration Fu = (Ftu )0≤t≤T on (u , Fu , Pu ) is defined by setting Ftu = σ ({us : 0 ≤ s ≤ t}) for every 0 ≤ t ≤ T . We shall also assume that VT is a weakly compact in distribution set of C([0, T ], Rd+3 )-random variables such that for every u ∈ VT there exists an m-dimensional Fu -Brownian motion B u = (Btu )0≤t≤T on (u , Fu , Fu , Pu ). In such a case, for given measurable functions f and g satisfying conditions (H ), x ∈ D, and every u ∈ VT there exists (see [83], Th.11.2.1 of Chap. 11) exactly one strong solution Vu of the x (V ) we denote a set initial valued problem (9.3.1). Similarly as above, by ZD T x x {V : (PF , B, V ) : WD (VT )}, where WD (VT ) is the set of all weak solutions (PFu u , B u , Vu ) to the initial value problem (9.3.1) corresponding to every u ∈ VT . From Theorem 8.1.2 of Chapter 8 the following result follows. Theorem 9.3.2 Let D be a nonempty bounded domain of R, and let U be a nonempty closed subset of Rd+3 . Assume that f and g are measurable and satisfy conditions (H) of Chapter 8, and let VT be a weakly compact in distribution set of C([0, T ], Rd+3 )-random variables such that ut (ω) ∈ U for every u ∈ VT , a.e. ω ∈ u and 0 ≤ t ≤ T , where ut = (πt , ϕ, ct ), π = (π 0 , π 1 , . . . , π d ), φ = (φt )0≤t≤T , and c = (ct )0≤t≤T . If K : R+ × Rd → R and  : R+ × Rd × U → R are continuous and bounded, then for every x ∈ D there exist u ∈ VT and x x V u, V u ∈ ZD (VT ) such that JD ( u ) = sup{JD (u, Vu ) : (u, Vu ) ∈ VT × ZD (VT )}.   The above theorem can be applied to some optimal control problems of the financial mathematics. In particular, Theorem 9.3.1 implies the existence of solution of an optimal portfolio selection problem presented in [83, Example 11.2.5, Chap. 11]. In a particular, it leads to the definition of such optimal selector and to the estimation of the value of considered portfolios. Example 9.3.1 Let Xt denote the wealth of a person at time t ≥ 0. Suppose that the person has the choice of two different investments. The price p1 (t) at time t

264

9 Mathematical Finance Problems

one of the assets is assumed to satisfy the equation : dp1 (t)/dt = p1 (t)(a + c Wt ), where Wt denotes the white noise and, a, c > 0 are constants measuring the average relative rate of change of p and the size of the noise, respectively. It is well-known (see [83], Problem 5 of Chap1) that the last equality can be interpreted as the Itô stochastic differential equation dp1 = ap1 dt + cp1 dBt . The investment is called risky, since c > 0. We assume that the price p2 of the other asset satisfies a similar equation, but with no noise: dp2 = bp2 dt. This investment is called safe. So it is natural to assume that b < a. At each instant, the person can choose how big fraction u of his wealth he will invest in the risky asset, thereby investing the fraction 1 − u in the safe one. This gives the following stochastic differential equation for wealth Xt = Xtu : dXt = [au + b(1 − u)]Xt dt + cuXt dBt . Suppose that starting with the wealth X0 = x > 0 at time t = 0, the person wants to maximize the expected utility of the wealth at some future time T > 0. If we allow no borrowing, i.e., require X ≥ 0, and a utility function U (x) = x r for every x > 0, and 0 < r < 1 is given, the problem is to find a fraction 0 ≤ u∗ ≤ 1 such that ∗ E[U (XTu )] = sup{E[U (XTu )] : 0 ≤ u ≤ 1}, where T > 0 is a fixed future time. In [83, Example 11.2.5, Chap. 11] the problem was solved by Hamilton–Jacobi– Bellman equation (see [83], Theorem 11.2.1 of Chap. 11), and the optimal control u∗ = (a − b)/[c2 (1 − r)] has been obtained. It is clear that the existence of the ∗ above optimal pair (u∗ , Xu ) follows immediately from Theorem 9.3.1. It can be verified that it satisfies a second-order algebraic equation, which in some special cases implies the existence of the optimal control u∗ presented above, and admits an estimation of E[U (XTu∗ )]. Indeed, by Itô’s formula (see Theorem 1.8.5 of Chapter 1) for every fixed 0 ≤ u ≤ 1 one gets XTu = x exp{[(a−b)+b−(1/2)c2 u2 ]t +(cu)Bt }. Then E[U (Xtu )] = x r E t,x [exp{rt[(a − b)u + b − (1/2)c2 u2 ] + (cru)Bt }]. Let us note (see [83], Exercise 2.2 of Chap. 1) that  √ E[f (Bt )] = 1/ 2π t

+∞

−∞

f (x) exp(−x 2 /2t)dx

for every function f such that the last integral converges. Taking in particular, f (x) = exp{rt[(a − b)u + b − (1/2)c2 u2 ] + (cru)x} one gets E[exp{rt[(a−b)u+b−(1/2)c2 u2 ]+(cru)Bt }]=exp{−(1/2)rt[c2 (1−r)u2 −2(a−b)u−b]},

because  +∞ −∞

f (x) exp[−x 2 /(2t)]dx =

√

2tπ exp{b(rt − rtu) − (1/2)u[−2art + c2 (rt − r 2 t)u]}.

Then E[U (Xtu )] = x r exp{−(1/2)rt[c2 (1 − r)u2 − 2(a − b)u − b]}. Hence it follows that −2Aut = t[c2 (1 − r)u2 − 2(a − b)u − b] = 0, where Aut =

9.3 Finance Optimal Control Problems

265 ∗

(1/r) log{E[U (Xtu )]/x r }. Thus the optimal pair (Xu , u∗ ) satisfies the following second-order equation ∗

c2 (1 − r)(u∗ )2 − 2(a − b)u∗ + log{(E t,x [U (XTu )]/x r )2/(rT ) } − b = 0. ∗

Therefore, log{(E[U (XTu )]/x r )2/(rT ) } = −c2 (1 − r)(u∗ )2 + 2(a − b)u∗ + b, which ∗ implies that log{(E[U (XTu )]/x r )2/(rT ) } is equal to the maximum of the function g(u) = −c2 (1 − r)u2 + 2(a − b)u + b defined for u ∈ [0, 1]. It is attained at u∗ = (a − b)/c2 (1 − r) and the mean value of the optimal performance functional ∗ ∗ U (XTu ) is then defined by : E[U (XTu )] = x r exp[rT /2(b + 2(b − a)u∗ − c2 (1 − r)(u∗ )2 )]. In particular, if x = 100, r = 0.2, a=10, b=7, c=10.3, and T = 10, then ∗ u∗ = 0.0353473 and E[U (XTu )] = 3062.77.   From Theorem 9.3.2 we can obtain the existence of solutions of optimal control problem, that generalize the optimal investment–consumption problem with borrowing considered by W.H. Fleming and Th. Zariphopoulou in the paper [19]. To begin with let us recall the definition of the optimal control problem presented in the paper [19]. W.H. Fleming and Th. Zariphopoulou have considered a market with two assets : a bond and a stock. The price Pt0 of the bond evolves according to the equation (1) dPt0 = rPt0 dt. The price of the stock satisfies the stochastic differential equation (2) dPt = bPt dt + σ Pt dBt . In both cases the initial conditions : P00 = p0 and P0 = p a.s. are satisfied. It is assumed that σ > 0 and 0 < r ≤ R ≤ b. The investor consumes, trades, and borrows continuously in time. Denoting by φt the amount borrowed, and by πt0 and πt amounts, which the investor puts at time t in bonds and stocks, respectively, the net amount of current wealth Xt is given by Xt = πt0 + πt − φt . If, on other hand, the investor chooses at time t to consume an amount ct , then the evolution for the investor’s wealth is given by the equation dXt = rXt dt + [(b − r)πt − (R − r)φt ]dt − ct dt + σ πt dBt . It is considered with an initial condition X0 = x for a fixed x ∈ D, where D is a bounded domain of the real line. The control variables are : πt , ct , and φt with control constrains ct ≥ 0, πt ≥ 0, and φ ≥ 0 a.s. for t ≥ 0. Moreover, the wealth of the investor has to stay non-negative, i.e., Xt ≥ 0 a.s., for t ≥ 0. The control are stochastic processes and they are called admissible if they satisfy the above constrains. The objective of the investor is to maximize, over all admissible controls, the total expected  ∞ discounted utility coming from consumption. The following value functional E{ 0 e−βt K(ct )dt} has to be maximized, where K is the utility function and β is the discount factor. We shall now consider the finance market described by the stochastic differential equation 

t

z t = z0 + 0



t

f (τ, uτ , zτ )dτ +

g(τ, uτ , zτ )dBτ 0

(9.3.3)

266

9 Mathematical Finance Problems

with ut = (πt , ct , t ) and an initial distribution P z0−1 = μx , where f (t, zt , πt , ct , φt ) = rzt + (b − r)πt − (R − r)φt − ct , g(t, zt , πt , ct , φt ) = σt a.s. for t ≥ 0 and μx is a probability measure defined for a given x ∈ R in Remark 6.3.1 of Chapter 6. Example 9.3.2 Consider the financial market described by the stochastic differential equation (9.3.3), where (σt )t≥0 is a deterministic square integrable function with values in Rm . Let U be a nonempty compact subset of the space R3 and F : R+ × D → Cl(R) be the multifunction defined by F (t, y) = {f (t, y, u) : u ∈ U } for every (t, y) ∈ R+ × D. It is clear (see [48], Lemma 2.1 of Chap. 7) that F is continuous and bounded. Therefore, by Theorem 8.2.1 of Chapter 8, for every x ∈ D, and continuous bounded functions K : R+ ×R → R and  : R+ ×R → R, , B, X) to a stochastic optimal control problem there is an optimal weak solution (P F described by the stochastic inclusion of the form   ⎧   ⎨X s ∈ cl t F (τ, X τ )dτ + t στ d B τ t − X s s ⎩ X0 = x, = sup{J x (X) : satisfied a.s. for every 0 ≤ s < t < ∞, and such that JDx (X) D / X 0 τD x (F, G)}, where J x (X) = E X , X ) . By remarks (t, X )dt + K(τ X ∈ ZD X t τD D D s can be assumed presented in Chapter 8 (see Notes and Remarks to Chapter 8), X to be a continuous stochastic process that has not to be F-adapted. Therefore, by virtue of Theorem 4.5.1 of Chapter 4, for every T > 0 there is a measurable selector   such that X t = x + t ϕτ dτ + t στ d B τ a.s. for 0 ≤ t ≤ T . ϕ = (ϕt )0≤t≤T for F ◦ X 0 0 t ( By the definition of the multifunction F we have ϕt ( ω) ∈ {f (t, X ω), u) : u ∈ U } for a.e. (t, ω) ∈ [0, T ] × , which can be written in the form ϕt ( ω) ∈ f (t, t ( ω)) t ( be , where t ( ω ) = {X ω)} × U . Let  ⊂ [0, T ] ×  for a.e. (t, ω) ∈ [0, T ] ×  the Lebesgue of measure zero set such that ϕt ( ω) ∈ f (t, t ( ω)) for every (t, ω) ∈ ) \ . Let (λt )t≥0 be a measurable stochastic process, a measurable ([0, T ] ×  selector of the set-valued mapping defined by t ( ω) = U for every (t, ω) ∈ t ( , such that (X . Put [0, T ] ×  ω), λt ( ω)) ∈ t ( ω) for every (t, ω) ∈ [0, T ] ×  t ( ) \  and ϕt ( ω) = ϕt ( ω) for (t, ω) ∈ ([0, T ] ×  ϕt ( ω ) = f (X ω), λt ( ω) for (t, ω) ∈ . It is clear that ϕ : [0, T ] ×  → R is a measurable function such . Then by Theorem 2.2.4 that ϕt ( ω) ∈ f (t, t ( ω)) for every (t, ω) ∈ [0, T ] ×  → R3 such that of Chapter 2, there exists a measurable function u : [0, T ] ×  . Let ω) = f (t, Xt ( ω), u(t, ω)) for every (t, ω) ∈ [0, T ] ×  u(t, ω) ∈ U and ϕt ( ut ( ω) = ut ( ω) for (t, ω) ∈ ([0, T ] × ) \  and ut ( ω) = λt ( ω) for (t, ω) ∈ . It t ( is clear that u is a measurable selector of such that ϕt ( ω) = f (t, X ω), ut ( ω)) \ , which by the definition of ω) = for every (t, ω) ∈ [0, T ] ×  ϕ implies that ϕt ( t ( . Therefore, there exists a measurable f (t, X ω), ut ( ω)) a.e. (t, ω) ∈ [0, T ] ×  t  τ , uτ ) + t στ dBτ , P selector of multifunction such that Xt = x + 0 f (τ, X 0 a.s. for every 0 ≤ t ≤ T . Then there exists an optimal pair (X, u) to the optimal control problem to the dynamical system described by the stochastic differential

9.3 Finance Optimal Control Problems

267

equation (9.3.3) with ut = (πt , ct , φt ) and an initial distribution P z0−1 = μx , where f (t, zt , πt , ct , φt ) = rzt + (b − r)πt − (R − r)φt − ct , g(t, zt , πt , ct , φt ) = σt a.s. for t ≥ 0 with a probability measure μx defined for a given x ∈ R in Remark 6.3.1 of Chapter 6.   For given measurable functions f : R+ × R+ × U → R and g : R+ × R+ × U → R satisfying conditions (H) of Chapter 8, and a nonempty closed set U ⊂ Rk we consider now a set-valued functional inclusions SFt (Xt ) − SFt (Xs ) ⊂ decFt Js,t (SF (F ◦ X)) + Js,t (SF (G ◦ X))

(9.3.4)

with F (t, x) = {f (t, x, z) : z ∈ U }, G(t, x) = {g(t, x, z) : z ∈ U }, F (t, A) = co F (t, A) and G (t, A) = co G(t, A) for (t, x) ∈ [0, T ] × Rd , and (t, A) ∈ [0, T ] × Conv(Rd ), respectively. Similarly as above, for given a complete filtered probability space PF = (, FT , F, P ) with a filtration F = (Ft )0≤t≤T satisfying the usual conditions, a real-valued F-Brownian motion B = (Bt )0≤t≤T defined on PF , a nonempty compact set M ⊂ R+ and T > 0, by SI(F, G, M, T ) we still denote sets of all strong solutions to the above setvalued functional inclusions (9.3.4) satisfying the initial condition X0 = co M. We shall show that for every X ∈ SI(F, G, M, T ) and a Lipschitz continuous utility function K : R+ → R+ , there exists a point τ ∈ [0, T ] such that Z(X)= :inf0≤t≤T {supx∈SFt (Xt ) E[K(x)]} is equal to E[K(xτ )], where xτ is an Fτ measurable selector of Xτ . For simplicity the set SFt (Xt ) will be denoted by St (X). We begin with the following lemma Lemma 9.3.1 Let F : R+ × Rd → Cl(Rd ) and G : R+ × Rd → Cl(Rd×m ) satisfy conditions (i) of the assumptions (H) of Chapter 7. Then for given a filtered probability space PF = (, F, F, P ), an m-dimensional F-Brownian motion B = (Bt )0≤t≤T , X ∈ SI(F, G, M, T ), and a Lipschitz continuous utility function K : Rd → R+ , the function γ : [0, T ]  t → supx∈St (Xt ) E[K(x)] is uniformly continuous. Proof Let X ∈ SI(F, G, M, T ) be fixed. By uniform mean value continuity of X, for every ε > 0 there is δε > 0 such that for every t1 , t2 ∈ [0, T ] such that |t1 − t2 | < δε one has Eh2 (Xt1 , Xt2 ) ≤ (ε/L)2 , where L > 0 is a Lipschitz constant of K. By the definition of function γ for every 0 ≤ t ≤ T we have γ (t) = sup{E[K(x)] : x ∈ SFt (Xt )}, which by Remark 2.3.2 of Chapter 2 implies that γ (t) = E[h(K[Xt ], {0})] for 0 ≤ t ≤ T . Therefore, for every t1 , t2 ∈ [0, T ] one has |γ (t1 ) − γ (t2 )| = |E[h(K[Xt1 ], {0})] − E[h(K[Xt2 ], {0})]| ≤ E|h(K[Xt1 ], {0}) − h(K[Xt2 ], {0})| ≤ Eh(K[Xt1 ], K[Xt2 ]), because for every A, B ∈ Cl(Rd ) one has |h(A, {0}) − h(B, {0})| ≤ h(A, B). By virtue of Lemma 7.4.1 of Chapter 7 we have E[h(K[Xt1 ], K[Xt2 ])] ≤

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LE[h(Xt1 , Xt2 )]. Hence, by H¨older’s inequality it follows 5 |γ (t1 ) − γ (t2 )| ≤ L E[h2 (Xt1 , Xt2 )] for every t1 , t2 ∈ [0, T ]. Therefore, |γ (t1 ) − γ (t2 )| ≤ ε for every t1 , t2 ∈ [0, T ] such   that |t1 − t2 | ≤ δε . Theorem 9.3.3 If the assumptions of Lemma 9.3.1 are satisfied, then for every X ∈ SI(F, G, M, T ) and a linear function K : Rd → R+ there exist τ ∈ [0, T ] and an Fτ -measurable selector xτ of Xτ such that Z(X) = E[K(xτ )]. Proof Let X ∈ SI(F, G, M, T ) be fixed. By Lemma 9.3.1, there exists τ ∈ [0, T ] such that Z(X) = sup{E[K(x)] : x ∈ Sτ (X)}. By Lemma 7.5.1 of Chapter 7, Sτ (X) is a bounded subset of the space L2 (, Fτ , Rd ). Therefore, there exists a sequence (un )∞ n=1 of Sτ (X) such that Z(X) = limn→∞ E[K(un )]. It is clear that Sτ (X) is a weakly compact subset of the space L2 (, Fτ , Rd ). Then there exists u ∈ Sτ (X) such that an appropriate subsequence of (un )∞ n=1 converges weakly to u. For simplicity, assume that un  u as n → ∞. By Dunford–Schwartz theorem, hence it follows that K(un )  K(un ) as n → ∞. Thus, E[K(un )] → E[K(u)] as n → ∞. Then Z(X) = E[K(u)]. But, u ∈ Sτ (X) implies the existence of an Fτ measurable selector xτ of Xτ such that xτ = u a.s. Thus, Z(X) = E[K(xτ )].   It is natural to ask if it is possible to obtain the above result if the function γ in Lemma 9.3.1 would be defined by setting γ : [0, T ]  t → supx∈decFt [cot ] E[K(x)]}, where t is the attainable set to SF I (F, G)? Unfortunately, the answer to this question is negative, because H-continuity of a set-valued mapping [0, T ] → t ⊂ L2 (, Ft , Rd ) does not imply continuity of such defined function γ . It follows from the following example. Example 9.3.3 Let (, F, P ) = ([0, 1], β, m), where β is a Borel σ -algebra on [0, 1] and m : β → [0, 1] is the Lebesgue measure on β. Consider a metric space Cl(Lp ([0, 1], R), H ), with 1 ≤ p < ∞ and the Hausdorff metric H . Let F : Lp ([0, 1], R) → Lp ([0, 1], R) be defined by setting F (A) = decβ (A) for every A ⊂ Lp ([0, 1], R). The multifunction F is not continuous with respect to metric H , p because there exists a sequence (An )∞ n=1 of subset of the space Cl(L ([0, 1], R), H ) p convergent to A ∈ Cl(L ([0, 1], R), H ) and such that H (F (An ), F (A)) does not tend to 0. Indeed, let πn = {0 < 1/n < 2/n < . . . < k/n < 1} be for every n = 1, 2, . . . a partition of the interval [0, 1] with k = 0, 1, 2, . . . , n, and let (fk ) be a sequence of functions f k : [0, 1] → [0, 1] defined by fk (t) = 1[k/n,(k+1)/n] for k = 0, 1, 2, . . . , n. Define now a sequence (An )∞ n=1 of subsets of the space Cl(Lp ([0, 1], R), H ) by setting An = {fk : k = 0, 1, . . . , n − 1} for n ≥ 1. For every k = 0, 1, 2, . . . , n we have 0 < fk (t) ≤ 1. Then all sets An are integrably bounded with the same majorant g(t) = 1. Moreover, fk p = 1/n1/p for every k = 0, 1, . . .. Therefore, the sequence (An )∞ n=1 converge with respect to the metric H to the set A = {0} ∈ Cl(Lp ([0, 1], R), H ). On the other hand, F (A) = decβ ({0}) = {0} while a set F (An ) = decβ (An ) contains a

9.3 Finance Optimal Control Problems

269

n−1 function f (t) = 1[k/n,(k+1)/n] (t) · fk (t) = 1. Since f p = 1, then 0 ¯ (An ), F (A)) ≥ h(f, ¯ h(F F (A)) = f − 0p = 1. It means that F (A) = decβ (A) is not continuous in a space Lp (, F, P ) for any 1 ≤ p < ∞.   Let us consider now a stochastic differential inclusion t t  Vt − Vs ∈ s F (τ, Vτ )dτ + s co (G ◦ V )dBτ a.s. for t ∈ [0, T ] . V0 ∈ M a.s. For given a multifunction F , and a family G = {g n : n ≥ 1}, satisfying the assumptions of Theorem 6.1.1 of Chapter 6, given a filtered probability space PF = (, F, F, P ) with filtration F = (Ft )0≤t≤T satisfying the usual conditions, an m-dimensional F-Brownian motion B = (Bt )0≤t≤T defined on PF , T > 0, and a nonempty compact set M ⊂ Rd , by S(F, G, M, T ), similarly as in Chapter 6, we still denote the set of all strong solution on the interval [0, T ] to the above initial value problem. By the above notations we obtain the following result. Theorem 9.3.4 If the assumptions of Theorem 6.1.1 of Chapter 6 are satisfied, then for a filtered probability space PF = (, F, F, P ) with filtration F = (Ft )0≤t≤T satisfying the usual conditions, an m-dimensional F-Brownian motion B = (Bt )0≤t≤T defined on PF , T > 0, a nonempty compact set M ⊂ Rd , and a linear utility function K : Rd → R+ there exists a sequence (tn )∞ n=1 ⊂ [0, T ] convergent to τ ∈ [0, T ] such that supx∈Zτ E[K(x)] ≤ sup{E[K(x)] : x ∈ Li Ztn }, where Zt = et (S(F, G, M, T ), and Li Ztn is the Kuratowski lower limit of a sequence (Ztn )∞ n=1 . Furthermore, for every n ≥ 1 there exists a sequence ⊂ S(F, G, M, T ) such that v(M) = limn→∞ {limk→∞ E[K(Vkn )]}, where (Vkn )∞ k=1 v(M) = inf0≤t≤T {supx∈Zt E[K(x)]}. Proof By virtue of Lemma 7.5.2 of Chapter 7, a set-valued mapping [0, T ]  t → Zt ⊂ L2 (, FT , Rd ) is l.s.c. Therefore, by [27, Prop.2.5 of Chap. 1] we have Zτ ⊂ Li Ztn . Then supx∈Zτ E[K(x)] ≤ sup{E[K(x)] : x ∈ Li Ztn }. By Lemma 7.5.1 of Chapter 7, EZt  ≤ α exp(βT ) < ∞. Therefore, inf0≤t≤T {supx∈Zt E[K(x)]} < ∞. Then there exists a sequence (tn )∞ n=1 ⊂ [0, T ] such that inf0≤t≤T {supx∈Zt E[K(x)]} = limn→∞ {supx∈Ztn E[K(x)]}. By the compactness of the interval [0, T ] there exists a subsequence of a sequence (tn )∞ n=1 convergent to τ ∈ [0, T ]. For simplicity assume that tn → τ as n → ∞. But, supx∈Z E[K(x)] = H¯ (K(Ztn ), {0}) for every n ≥ 1, where H¯ tn

is the Hausdorff sub-distance on the space Cl(L2 (, FT , Rd )). Then v(M) = limn→∞ H¯ (K(Ztn ), {0}). Hence, similarly as above it follows that for every n ≥ 1 ¯ there exists a sequence (unk )∞ k=1 of the set K(Ztn ) such that H (K(Ztn ), {0}) = n 2 limk→∞ E|uk | . On the other hand, similarly as in the Example 9.3.2, we can see that for every n ≥ 1 there exists a sequence (Vkn )∞ k=1 ⊂ S(F, G, M, T ) such that unk = K(Vkn (tn )) a.s. Then v(M) = limn→∞ {limk→∞ E[K(Vkn (tn ))]}.  

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9.4 Recursive Utility Optimization Problem Apart from the classical initial value problems described by Itô stochastic differential equations, a lot of applications deal with stochastic differential equations with terminal conditions. In particular, such equations occur in financial market, option pricing, optimal investment strategy, stochastic differential utility, and recursion functions. Since Itô stochastic differential equations with terminal conditions usually have no adapted solutions, many attempts have been made to overcome this difficulty. E. Pardoux and S.G. Peng, and D. Duffi and L. Epstein (see [84], and [14]) proposed a special kind of stochastic differential equations with a terminal condition, called the backward stochastic differential equation, of the form 

T

Xt = ξ + t



T

f (τ, Xτ , Yτ )dτ −

Yτ dBτ ,

(9.4.1)

t

where ξ ∈ Rn . The solution to such equation is a pair of an appropriate pair (X, Y ) processes satisfying (9.4.1) a.s. for every 0 ≤ t ≤ T . Duffie and Epstein introduced (see [14]) so-called stochastic differential utility as a solution of the T following stochastic equation Yt = E[Y (T ) + t f (τ, Yτ )dτ |Ft ]. For given ξ ∈ R, a function f : [0, T ] × R × Rd → R, and a probability measure μ on a Borel σ -algebra β(Rd ), we shall consider an optimization problem for recursive utility processes defined by weak solutions of backward stochastic differential equations of the form (9.4.1). By a weak solution to (9.4.1) we mean a system (PF , X, Y, B) consisting of a complete filtered probability space PF = (, F, F, P ) with a filtration F = (Ft )0≤t≤T satisfying the usual conditions, a realvalued continuous process X = (Xt )0≤t≤T , d-dimensional continuous F-adapted stochastic process Y = (Yt )0≤t≤T such that E[sup0≤t≤T |Yt |2 ] < +∞, and a d-dimensional F-Brownian motion B = (Bt )0≤t≤T defined on PF satisfying the following conditions: 

t t Xt − Xs = s f (τ, Xτ , Yτ )dτ − s Yτ dBτ , a.s. for 0 ≤ s < t ≤ T , P XT−1 = μξ , P Y0−1 = μ, (9.4.2)

where μξ is a probability measure on β(Rd ) defined similarly as μx in Remark 6.3.1 of Chapter 6. In what follows, the set of all weak solutions to the above initialterminal problem (9.4.2) will be denoted by X (f, ξ, μ). For a given above ξ ∈ R, a function f : [0, T ] × R × Rd → R, and a probability measure μ on β(Rd ) we shall , X, Y , B) to (9.4.2) such that J (X) = sup{J (X) : look for a weak solution (P F X ∈ Z(f, ξ, μ)}, where Z(f, ξ, μ) = {X : (PF , X, Y, B) ∈ X (f, ξ, μ)}, and J (X) = E[X0 ] for every X ∈ Z(f, ξ, μ). In what follows, the above optimization problem will be denoted by

9.4 Recursive Utility Optimization Problem

271

⎧ t t ⎪ ⎨ Xt − Xs = s f (τ, Xτ , Yτ )dτ − s Yτ dBτ , a.s. for 0 ≤ s < t ≤ T , P XT−1 = μξ , P Y0−1 = μ, ⎪ Z (f,ξ,μ) ⎩ J (X) −→ max . (9.4.3) Similarly to the proof of Theorem 7.2.1 of Chapter 7 we can prove the following existence theorem. Theorem 9.4.1 If f : [0, T ] × R × Rd → R is continuous bounded, then X (f, ξ, μ) = ∅ for every ξ ∈ R and every probability measure μ on β(Rd ). Proof (Sketch Proof) Let PF = (, F, F, P ) be a complete filtered probability space with a filtration F = (Ft )0≤t≤T satisfying the usual conditions such that there exists a d-dimensional F-Brownian motion B = (Bt )0≤t≤T defined on PF . Let Y = (Yt )0≤t≤T be d-dimensional continuous F-adapted stochastic process defined on PF such that E[sup0≤t≤T |Yt |2 ] < +∞. Define on PF a sequence (Xn )∞ Xn = (Xtn )0≤t≤T such that Xtn = ξ a.s. for n=1 of continuous processes  T T t ∈ [T , 2T ] and Xtn = ξ + t f (τ, Xτn+T /n , Yτ )dτ − t Yτ dBτ a.s. for 0 ≤ t ≤ T and n = 1, 2, . . .. Let us observe that for every n ≥ 1 the process Xn is defined step by step starting with the interval [T − T /n, T ]. For example, for T T t ∈ [T − T /n, T ] we have Xtn = ξ + t f (τ, ξ, Yτ )dτ − t Yτ dBτ a.s. For  T T n t ∈ [T − 2T /n, T − T /n] we have Xtn = ξ + t f (τ, X τ +T /n , Yτ )dτ − t Yτ dBτ   tn = ξ + T f (τ, ξ, Yτ )dτ − T Yτ dBτ a.s. Similarly to the proof a.s. with X t t of Theorem 7.2.1 of Chapter 7 we can verify that a sequence (Xn )∞ n=1 is tight. Hence it follows that also a sequence {(Xn , Y, B)}∞ is tight. Then there exist n=1 ∞ ∞ k k a probability space (, F, P ), sequences (X )k=1 and (Y )k=1 of continuous k = (X tk )0≤t≤T and Y k = (Y tk )0≤t≤T with X tk :  → R stochastic processes X ∞ k d k k = and Yt :  → R , a sequence (B )k=1 of d-dimensional Brownian motions B k t )0≤t≤T , continuous stochastic processes X = (X t )0≤t≤T and Y = (Y t )0≤t≤T (B t :  t :  → R and Y → Rd , and d-dimensional Brownian motions with X d k k k = (B tk )0≤t≤T such that (Xk , Y, B) = (X , Y , B ) for every k ≥ 1, and such that B k , Y k , B k )}∞ converges a.s. uniformly in t ∈ [0, T ] to (X, Y , B). the sequence {(X k=1 Hence, similarly to the proof of Theorem 7.2.1 of Chapter 7 it follows that the , X, Y , B), with a filtration , is a weak solution system (P F defined by a process Y F to (9.4.2).   Similarly to the proof of Theorem 7.3.1 of Chapter 7, we also get the following result. Theorem 9.4.2 If f : [0, T ] × R × Rd → R is continuous bounded, then the set Z(f, ξ, μ) = {X : (PF , X, Y, B) ∈ X (f, ξ, μ)} is weakly compact in distribution for every ξ ∈ R and every probability measure μ on β(Rd ).   We can prove now the existence of solution to the recursive utility optimization problem (9.4.3). It will be followed from the following lemma.

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Lemma 9.4.1 Let f : [0, T ] × R × Rd → R and g : R → R be continuous bounded functions. Then for every ξ ∈ R and every probability measure μ on β(Rd ) , X, Y , B) ∈ X (F, ξ, μ) such that E[g( X 0 )] = sup{E[g(X0 )] : X ∈ there is (P F Z(f, ξ, μ)}. Proof By boundedness of a function g : R → R we have α = : sup{E[g(X0 )] : (PF , X, Y, B) ∈ X (F, ξ, μ)} < ∞. Then there exists a sequence (Xn )∞ n=1 ⊂ Z(f, ξ, μ) such that α = limn→∞ En [g(X0n )]. By weak compactness in distribution of Z(F, ξ, μ) there exists (PF , X, Y, B) ∈ X (F, ξ, μ) and a sequence n n n n ∞ {(PFk k , Xk , Y k , B k )}∞ k=1 , a subsequence of the sequence {(PFn , X , Y , B )}n=1 , k k k such that (X , Y , B ) ⇒ (X, Y, B) as k → ∞. By Corollary 1.5.3 of Chapter 1 P ) and random variables X k , Y k , X, and Y , F, there exist a probability space ( d , P ) with values at C([0, T ], R) and C([0, T ], R ), respectively, such that , F on ( d k k d tk − X t | + (Xk , Y k ) = (X , Y ) and (X, Y ) = (X, Y ) and such that sup0≤t≤T [|X k t ] → 0 a.s. as k → ∞. Furthermore, there exists a d-dimensional Brownian t −Y Y P ) such that (P , X, = (B t )0≤t≤T on ( Y , B) ∈ X (f, ξ, μ) with , F, motion B F = ( , ), where , B). , F P F, P F is a natural filtration defined by a process (Y F X k )] and limk→∞ E[g( X k )] = But α = limk→∞ Ek [g(X0k )], Ek [g(X0k )] = E[g( 0 0 X 0 )]. X 0 )]. Then α = E[g(   E[g( We obtain now the existence of the solution of the recursive utility optimization problem (9.4.3). Theorem 9.4.3 Let f : [0, T ] × R × Rd → R be continuous bounded function. Then for every ξ ∈ R, and every probability measure μ on β(Rd ) there is , X, Y , B) ∈ X (F, ξ, μ) such that E[ X 0 ] = sup{E[X0 ] : X ∈ Z(f, ξ, μ)}. (P F Proof Let us note that for every (PF , X, Y, B) ∈ X (F, ξ, μ) one has E|X0 |2 ≤ λ, T where λ = 3|ξ |2 + 3T L + 0 EYτ 2 dτ with L > 0 such that |f (t, x, y)| ≤ L for every (t, x, y) ∈ [0, T ] × R × Rd . Then X0 ∈ Bλ a.s. for every (PF , X, Y, B) ∈ X (F, ξ, μ), where Bλ = [−λ, λ]. Let I : Bλ → R be the identity mapping and let g : R → R be its continuous extension on the whole real line. We have g (x) = x for x ∈ Bλ and g is continuous bounded. The result follows now immediately from Lemma 9.4.1 applied to a function g defined above, because g is continuous bounded and such that g (X0 ) = X0 a.s. for every (PF , X, Y, B) ∈ X (F, ξ, μ).   In practical applications, recursive utility optimization problems are considered ξ with a functional J defined by J (X, ξ ) = X0 depending on a terminal value ξ ∈ U , where Xξ is a solution to the recursive utility stochastic differential equation corresponding to a terminal value ξ ∈ U . In such case the recursive utility optimization problem is considered as some optimal control problem with control parameters ξ ∈ U . Unfortunately, by the method connected with weak solutions of recursive utility stochastic equations, we have to consider the above type of optimal ξ control problems with a functional J of the form J (X, ξ ) = E[X0 ]. With such defined functional J we can get a solution to the above described optimal control problem in a similarly way as it was presented in the proof of Theorem 8.1.1 of Chapter 8.

9.5 Notes and Remarks

273

Recursive utility optimization problems can be described by a much general backward stochastic differential equations. For example, much more general backward stochastic equation is considered in the paper [8]. It was defined for given functions f : [0, T ] × D(Rd ) × D(Rm ) → Rd and H : D(Rm ) → Rd in [8] as a relation of the form :  

 T  (9.4.4) Xt = E H (Y ) + f (τ, X, Y )dτ Ft . t

Given a probability measure μ on a Borel σ -algebra of the Skorokhod space D(Rd ) of càdlàg processes on [0,T], a weak solution to (9.4.4) was defined in [8] as a system (PF , X, Y, ) consisting of a complete filtered probability space PF = (, F, F, P ) with a filtration F = (Ft )0≤t≤T satisfying the usual conditions, a pair (X, Y ) ∈ D(Rm ) × D(Rd ) of càdlàg processes such that (i) (ii) (iii) (iv) (iv)

P Y −1 = μ, Y is F-adapted, T E|H (Y )| < ∞, and E 0 |f (τ, X, Y )|dτ < ∞, every FY -martingale is also F-martingale, T Xt = E[H (Y ) + t f (τ, X, Y )dτ |Ft ] a.s. for 0 ≤ t ≤ T .

To the best knowledge of the author’s up to now the recursive utility optimization problem for systems described by the above backward stochastic differential equations was not considered. The problem of the weak compactness in distribution of the set of all weak solutions to such equations is also still open problem. But, there is a subset of the set of all weak solutions to backward stochastic differential equations of the form (9.4.4) consisting of filtered probability spaces with continuous filtration. Such weak solutions to (9.4.4) can be called as continuous. It can be verified, similarly as in the case of stochastic differential inclusions, that the set of all continuous weak solutions to (9.4.4) is weakly compact in distribution. Therefore, the recursive utility optimization problems for systems described by continuous weak solutions to (9.4.4) can be also considered. In this case we can also consider the problem for systems described by continuous  t weak solutions to backward stochastic differential inclusions : xs ∈ E[xt + s F (τ, xτ )|Fs ] with a terminal condition xT ∈ H (xT ) considered in the author’s paper [46]. It can be proved (see [46], Th.6.2) that the set X (F, H ) of all continuous weak solutions to such terminal value problem is weakly compact in distribution.

9.5 Notes and Remarks Results of this chapter contain some examples concern with applications of the results on stochastic optimal control problems presented in Chapter 8 and properties of solutions sets of stochastic functional and set-valued functional inclusions, to some finance mathematics problems. Introductory notions of the financial

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mathematics presented in Section 1 of the chapter, come from B. Øksendal [83]. The results of Section 2 and Section 3 have not been published anywhere. The comprehensive approach to the financial mathematics can be found in the monograph of A.N. Shiryaev [89]. Apart from the theory of the financial mathematics, selected notions of the theory of stochastic processes are also presented there. To explain the specific nature of financial problems the author of the monograph [89] presents the basing objects and structures of financial theory. It is illustrated by a chart showing connections between the theory of finance and its practice. In the financial market the following financial instruments can be distinguished: underlying instruments, called the primary instruments, and derivative instruments, called the secondary ones. To the first of the above group belong : bank accounts, bonds, stocks and the second one contains : options, future contracts, warrants, swaps, spreads. The financial activities can be described in terms of the dilemma : consumption–investment. Let us note that the recursive utility optimization problem is connected with recursive economies based on a paradigm of individuals making a series of two-period optimization decisions over time. Let us recall differences between recursive and neoclassical paradigms. The neoclassical model assumes a one-period utility maximization for a consumer and one-period profit maximization by a producer. A time-series path in the neoclassical model is a series of these oneperiod utility maximization. In contrast, a recursive model involves two or more periods, in which the consumer or producer trades off benefits and costs across the two time periods. A time-series path in the recursive model is the result of a series of these two-period decisions. In the neoclassical model, the consumer or producer maximizes utility (or profits). In the recursive model, the subject maximizes value or welfare, which is the sum of current rewards or benefits and discounted future expected value.

References

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Index

A Affine mapping, 6 Alaoglu theorem, 11 Alexiewicz, A., 58, 105 American option, 262 Approximate continuity, 113 Ascoli theorem, 17 Aubin, J.P., 79 Aumann integral, 107 Aumann, R., 139

B Backward stochastic differential equation, 270 Banach space, 5 Billingsley, P., 58 Bochner integrable, 21 Bochner integral, 22 Boc¸san, G., 193 Borel-Cantelli lemma, 24 Brandao, A.I., 247 Brownian motion, 40

C Càdlàg process, 37 Canonical embedding mapping, 11 Carathèodory selector, 72 Cauchy sequence, 5 Cellina, A., 79 Chebychev’s inequality, 26 Claster point, 15 Closed convex hull, 8

Closed decomposable hull, 73 Compact set, 3 Complete measure, 24 Conditional expectation, 100 Conditional quadratic variation, 55 Consumption process, 261 Contingent claim, 262 Continuous process, 37 Convergent in measure, 18 Convex hull, 8 Convex set, 6 Cylinder sets, 38

D De Blasi, F., 247 Decomposable hull, 73 Distel, J., 58 Distribution of random variable, 27 Doob inequality, 40 Duality brackets, 7 Dual pair, 7 Dunford, N., 58, 105 Dunford-Schwartz theorem, 7

E ˘ Eberlein-Smulian theorem, 10 Evaluation mapping, 38 Exercise price, 261 Exit time, 42 Extension of filtered probability space, 52 Extension of probability space, 52

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280 F F-adapted process, 37 Fair price, 262 Fatou lemma, 21 F-Brownian motion, 41 Fernique theorem, 28 Filtration, 24 Fleming, W.H., 265 F-Markov’s process, 39 F-martingale, 39 F-non-anticipative process, 37 F-optional process, 37 F-predictable process, 37 F-progressively measurable process, 37 Frankowska, H., 79 Friedman, A., 58 Fryszkowski, A., 79, 105, 139 F-stopping time, 36 Fubini theorem, 85 G Geometric Brownian motion, 52 Graph of multifunction, 61 Graph measurable multifunction, 69 Gronwall lemma, 21 H Hausdorff distance, 12 Hausdorff space, 2 Hedging strategy, 262 Hess, Ch., 105, 131, 139 Hiai, F., vii, 105, 193 Hildenbrand, W., 79 H-lower semicontinuous, 62 H¨older inequality, 82 H-upper semicontinuous, 62 Hukuhara, M., 15, 68 Hukuhara derivative, 68 Hukuhara difference, 15 Hu, Sh., 58, 79, 79 I Iervolino, F., 247 Indecomposable sets, 86 Indefinite integral, 47 Indistinguishable process, 38 Itô process, 50 Itô’s stochastic integral, 44 J Jakod, J., 58 Jakubowski, A., 58

Index Jensen’s inequality, 40 Jung, E.J., 193 K Kallemberg, O., 58 Karatzas, I., 58 Kisielewicz, M., vii, 79, 193 Kim, J.H., 193 Klein, E., 79 ˘ Krein-Smulian theorem, 10 Kucia, A., 79 Kunita-Watanabe inequality, 55 Kuratowski and Ryll-Nardzewski Kuratowski, K., 15 L Lebesgue integrable, 19 Lebesgue integral, 19 Lebesgue point, 113 Lebesgue theorem, 21 Li, I., 138 Li, S., 138 Limit point, 15 Linear mapping, 6 Linear topological space, 5 Lipschitz continuous selection, 68 Locally convex space, 7 Local martingale, 40 Lower semicontinuous, 61 Lower topology, 61 Lyapunov theorem, 18 M Markov process, 39 Martingale, 39 Maturity date, 261 Mazur theorem, 9 Measurable multifunction, 69 Measurable process, 37 Measurable selection theorem, 69 Measurable selector, 69 Measurable space, 17 Measure of bounded variation, 18 Measure space, 17 Memin, J., 58 Michael’s continuous selection theorem, 65 Michta, M., 58, 193, 209 Minkowski inequality, 82 Modification of process, 38 Monotone convergence theorem, 20 Motyl, J., 209

Index N Normed space, 5 Norm reflexive space, 11 Norm topology, 5 Nowak, A., 79

O Øksendal, B., 58

P Papageorgiou, N.S., 58, 79 Parametrized set-valued function, 62 Payoff rate, 262 Polarization identity, 55 Polish space, 5 Portfolio, 260 Protter, Ph., 58 Przesławski, K., 79

Q Quadratic variation process, 55

281 Skorokhod, A.V., 56 Słonimski, L., 58 ˘ Smulian theorem, 11 Standard extension of probability space, 52 Steiner point map, 66 Stochastic differential, 50 Stochastic functional integral, 118 Stopped processes, 63 Stricker, C., 58 Strong measurable, 21 Subtrajectory integrals, 74 Sudakov-Slepian-Fernique theorem, 27 Supermartingale, submartingale, 39 Support function, 7

T Terminal payoff rate, 262 Thompson, A.C., 79 Tight sets, 29 Tolstonogov, A., 247 Topological (Kuratowski) limit, 16 Topological space, 2

R Radon-Nikodym Property (RNP), 23 Radon-Nikodym theorem, 19 Random variable, 26 Relatively compact set, 3 Recursive utility processes, 270 Reflexive space, 11 Relatively sequentially compact set, 3 Representation theorem, 52 Representation theorem for decomposable sets, 75 Rybi´nski, L., 79 Ryl-Nardzewski, C., 69

U Uhl, J., 58 Uhl theorem, 23 Umegaki, H., 79 105 Uniform operator topology, 6 Uniformly equicontinuous, 17 Upper semicontinuous, 61 Upper topology, 61 Usual conditions, 24

S Schwarz, J.T., 58, 105 Selector (selection) of multifunction, 63 Semimartingale, 40 Separable space, 3 Separation theorem, 8 Sequentially closed set, 2 Sequentially closure of a set, 2 Sequentially compact set, 3 Set-valued conditional expectation, 100 Set-valued martingale, 103 Shiryaev, A.N., 58 Shreve, S.A., 58

W Weak Cauchy sequence, 7 Weak convergence in distributions, 30 Weak limit of a sequence, 7 Weak∗ - topology, 11 Weakly closed, 8 Weakly compact, 8 Weakly measurable multifunction, 69

V Vitali theorem, 23

Z Zhang, J., 139