Advances in Mathematical Economics Volume 19 [2015 ed.] 4431554882, 9784431554882, 9784431554899

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Managing Editors Shigeo Kusuoka

Toru Maruyama

The University of Tokyo Tokyo, JAPAN

Keio University Tokyo, JAPAN

Editors Robert Anderson Jean-Michel Grandmont University of California, CREST-CNRS Berkeley Malakoff, FRANCE Berkeley, U.S.A.

Kunio Kawamata Keio University Tokyo, JAPAN

Hiroshi Matano Norimichi Hirano Charles Castaing The University of Tokyo Yokohama National Université Montpellier II Tokyo, JAPAN University Montpellier, FRANCE Yokohama, JAPAN Kazuo Nishimura Francis H. Clarke Kyoto University Université de Lyon I Kyoto, JAPAN Villeurbanne, FRANCE Tatsuro Ichiishi The Ohio State University Egbert Dierker Ohio, U.S.A. Yoichiro Takahashi University of Vienna The University of Tokyo Vienna, AUSTRIA Tokyo, JAPAN Alexander D. Ioffe Israel Institute of Darrell Duffie Akira Yamazaki Technology Stanford University Hitotsubashi University Haifa, ISRAEL Stanford, U.S.A. Tokyo, JAPAN Lawrence C. Evans University of California, Seiichi Iwamoto Makoto Yano Kyushu University Berkeley Kyoto University Fukuoka, JAPAN Berkeley, U.S.A. Kyoto, JAPAN Takao Fujimoto Fukuoka University Fukuoka, JAPAN

Kazuya Kamiya The University of Tokyo Tokyo, JAPAN

Aims and Scope. The project is to publish Advances in Mathematical Economics once a year under the auspices of the Research Center for Mathematical Economics. It is designed to bring together those mathematicians who are seriously interested in obtaining new challenging stimuli from economic theories and those economists who are seeking effective mathematical tools for their research. The scope of Advances in Mathematical Economics includes, but is not limited to, the following fields: – Economic theories in various fields based on rigorous mathematical reasoning. – Mathematical methods (e.g., analysis, algebra, geometry, probability) motivated by economic theories. – Mathematical results of potential relevance to economic theory. – Historical study of mathematical economics. Authors are asked to develop their original results as fully as possible and also to give a clear-cut expository overview of the problem under discussion. Consequently, we will also invite articles which might be considered too long for publication in journals.

Shigeo Kusuoka • Toru Maruyama Editors

Advances in Mathematical Economics Volume 19

123

Editors Shigeo Kusuoka Professor Emeritus The University of Tokyo Tokyo, Japan

Toru Maruyama Professor Emeritus Keio University Tokyo, Japan

ISSN 1866-2226 ISSN 1866-2234 (electronic) Advances in Mathematical Economics ISBN 978-4-431-55488-2 ISBN 978-4-431-55489-9 (eBook) DOI 10.1007/978-4-431-55489-9 Springer Tokyo Heidelberg New York Dordrecht London © Springer Japan 2015 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper

Contents

On the Integration of Fuzzy Level Sets . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Charles Castaing, Christiane Godet-Thobie, Thi Duyen Hoang, and P. Raynaud de Fitte

1

A Theory for Estimating Consumer’s Preference from Demand . . . . . . . . . . . Yuhki Hosoya

33

Least Square Regression Methods for Bermudan Derivatives and Systems of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Shigeo Kusuoka and Yusuke Morimoto Discrete Time Optimal Control Problems on Large Intervals . . . . . . . . . . . . . . Alexander J. Zaslavski

57 91

Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 137

v

Adv. Math. Econ. 19, 1–32 (2015)

On the Integration of Fuzzy Level Sets Charles Castaing, Christiane Godet-Thobie, Thi Duyen Hoang, and P. Raynaud de Fitte

Abstract We study the integration of fuzzy level sets associated with a fuzzy random variable when the underlying space is a separable Banach space or a weak star dual of a separable Banach space. In particular, the expectation and the conditional expectation of fuzzy level sets in this setting are presented. We prove the SLLN for pairwise independent identically distributed fuzzy convex compact valued level sets through the SLLN for pairwise independent identically distributed convex compact valued random set in separable Banach space. Some convergence results for a class of integrand martingale are also presented.

JEL Classification: C01, C02. Mathematics Subject Classification (2010): 28B20, 60G42, 46A17, 54A20. C. Castaing Département de Mathématiques, Case courrier 051, Université Montpellier II, 34095 Montpellier Cedex 5, France e-mail: [email protected] C. Godet-Thobie Laboratoire de Mathématiques de Bretagne Atlantique, Université de Brest, UMR CNRS 6295, 6, avenue Victor Le Gorgeu, CS 9387, F-29238 Brest Cedex3, France e-mail: [email protected] T.D. Hoang Quang Binh University, Quang Binh, Viet Nam e-mail: [email protected] P. Raynaud de Fitte () Laboratoire Raphaël Salem, UFR Sciences, Université de Rouen, UMR CNRS 6085, Avenue de l’Université, BP 12, 76801 Saint Etienne du Rouvray, France e-mail: [email protected] 1

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Keywords Conditional expectation • Fuzzy convex • Fuzzy martingale • Integrand martingale • Level set • Upper semicontinuous

Article type: Research Article Received: November 5, 2014 Revised: December 1, 2014

1 Introduction The study of fuzzy set-valued variables was initiated by Feron [10], Kruse [18], Kwakernaak [19, 20], Puri and Ralescu [25], Zadeh [30]. In particular, Puri and Ralescu [25] introduced the notion of fuzzy set valued random variables whose underlying space is the d -dimensional Euclidean space Rd . Concerning the convergence theory of fuzzy set-valued random variables and its applications we refer to [15, 21–23, 25–27]. In this paper we present a study of a class of random fuzzy variables whose underlying space is a separable Banach space E or a weak star dual Es of a separable Banach space. The paper is organized as follows. In Sect. 2 we summarize and state the needed measurable results in the weak star dual of a separable Banach space. In particular, we present the expectation and the conditional expectation of convex weak star compact valued Gelfand-integrable mappings. In Sect. 3 we present the properties of random fuzzy convex upper semi continuous integrands (variables) in Es . In Sect. 4, the fuzzy expectation and the fuzzy conditional expectation for random fuzzy convex upper semi continuous variables are provided in this setting. Section 5 is devoted to the SLLN for fuzzy convex compact (compact) valued random level sets through the SLLN for convex compact (compact) valued random sets. The above results lead to a new class of integrand martingales that we develop in Sect. 6. Some convergence results for integrand martingales are provided. Our paper provides several issues in Fuzzy set theory, but captures different tools from Probability and Set-Valued Analysis and shows the relations among them with a comprehensive concept.

2 Integration of Convex Weak Star Compact Sets in a Dual Space Throughout this paper, .; F ; P / is a complete probability space, E is a Banach space which we generally assume to be separable, unless otherwise stated, D1 D .ek /k2N is a dense sequence in the closed unit ball of E, E  is the topological

On the Integration of Fuzzy Level Sets

3

dual of E, and B E (resp. B E  ) is the closed unit ball of E (resp. E  ). We denote by cc.E/ (resp. cwk.E/) (resp. ck.E/) the set of nonempty closed convex (resp. weakly compact convex) (resp. compact convex) subsets of E. Given C 2 cc.E/, the support function associated with C is defined by ı  .x  ; C / D supf< x  ; y >; y 2 C g .x  2 E  /: We denote by dH the Hausdorff distance on cwk.E/. A cc.E/-valued mapping C W  ! cc.E/ is F -measurable if its graph belongs to F ˝ B.E/, where B.E/ is the Borel tribe of E. For any C 2 cc.E/, we set jC j D supfkxk W x 2 C g: 1 We denote by Lcwk.E/ .F / the space of all F -measurable cwk.E/-valued multifunctions X W  ! cwk.E/ such that ! ! jX.!/j is integrable. A sequence 1 .Xn /n2N in Lcwk.E/ .F / is bounded if the sequence .jXn j/n2N is bounded in L1R .F /. A F -measurable closed convex valued multifunction X W  ! cc.E/ is integrable if it admits an integrable selection, equivalently if d.0; X / is integrable. We denote by Es , (resp. Eb ), (resp. Ec ) the vector space E  endowed with the topology .E  ; E/ of pointwise convergence, alias w -topology (resp. the topology s  associated with the dual norm jj:jjEb ), (resp. the topology c of compact convergence) and by Em  the vector space E  endowed with the topology m D .E  ; H /, where H is the linear space of E generated by D1 , that is the Hausdorff locally convex topology defined by the sequence of semi-norms

Pn .x  / D maxfjhek ; x  ij W k  ng;

x  2 E  ; n 2 N:

Recall that the topology m is metrizable, for instance, by the metric dE  .x  ; y  / WD m

1 X 1 .jhek ; x  i  hek ; y  ij ^ 1/; x  ; y  2 E  : 2k kD1

We assume from now on that dE  is held fixed. Further, we have m  w  m c  s  : On the other hand, the restrictions of m , w , c to any bounded subset of E  coincide and the Borel tribes B.Es /, B.Em  / and B.Ec / associated with Es , Em  , Ec , are equal, but the consideration of the Borel tribe B.Eb / associated with the topology of Eb is irrelevant here. Noting that E  is the countable union of closed balls, we deduce that the space Es is a Lusin space, as well as the metrizable topological space Em  . Let K  D cwk.Es / be the set of all nonempty convex weak compact subsets in E  . A K  -valued multifunction (alias mapping for short) X W   Es is scalarly F -measurable if, 8x 2 E, the support function ı  .x; X.:// is F -measurable, hence its graph belongs to F ˝ B.Es /. Indeed, let .fk /k2N be a sequence in E which separates the points of E  , then we have

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x  2 X.!/ iff hfk ; x  i  ı  .fk ; X.!// for all k 2 N. Consequently, for any Borel set G 2 B.Es /, the set X  G D f! 2  W X.!/ \ G ¤ ;g is F -measurable, that is, X  G 2 F , this is a consequence of the Projection Theorem (see e.g. [8, Theorem III.23]) and of the equality X  G D proj fGr.X / \ .  G/g: In particular if u W  ! Es is a scalarly F -measurable mapping, that is, if for every x 2 E, the scalar function ! 7! hx; u.!/i is F -measurable, then the function f W .!; x  / 7! jjx   u.!/jjEb is F ˝ B.Es /-measurable, and for every fixed ! 2 ; f .!; :/ is lower semicontinuous on Es , i.e. f is a normal integrand. Indeed, we have jjx   u.!/jjEb D sup jhek ; x   u.!/ij: k2N

As each function .!; x  / 7! hek ; x   u.!/i is F ˝ B.Es /-measurable and continuous on Es for each ! 2 , it follows that f is a normal integrand. Consequently, the graph of u belongs to F ˝ B.Es /. Let B be a sub--algebra of F . It is easy and classical to see that a mapping u W  ! Es is .B; B.Es // measurable iff it is scalarly B-measurable. A mapping u W  ! Es is said to be scalarly integrable (alias Gelfand integrable), if, for every x 2 E, the scalar function ! 7! hx; u.!/i is F -measurable and integrable. We denote by GE1  ŒE.F / the space of all Gelfand integrable mappings and by L1E  ŒE.F / the subspace of all Gelfand integrable mappings u such that the function juj W ! 7! jju.!/jjEb is integrable. The measurability of juj follows easily from the above considerations. 1 1 More generally, by Gcwk.E  .; F ; P / (or Gcwk.E  / .F / for short) we denote the s s / space of all scalarly F -measurable and integrable cwk.Es /-valued mappings and 1 1 by Lcwk.E  .; F ; P / (or Lcwk.E  / .F / for short) we denote the subspace of s s /  all cwk.Es /-valued scalarly integrable and integrably bounded mappings X , that is, such that the function jX j W ! ! jX.!/j is integrable, here jX.!/j WD supy  2X.!/ jjy  jjEb , by the above consideration, it is easy to see that jX j is F measurable. 1 1 For any X 2 Lcwk.E  .F /, we denote by SX .F / the set of all Gelfands / integrable selections of X . The Aumann-Gelfand integral of X over a set A 2 F is defined by EŒ1A X  D

Z

A

X dP WD f

Z

A

f dP W f 2 SX1 .F /g:

On the Integration of Fuzzy Level Sets

5

Let B be a sub--algebra of F and let X be a K  -valued integrably bounded random set, let us define SX1 .B/ WD ff 2 L1E  ŒE.; B; P / W f .!/ 2 X.!/ a.s.g and the multivalued Aumann-Gelfand integral (shortly expectation) EŒX; B of X EŒX; B WD f

Z

f dP W f 2 SX1 .B/g:

As SX1 .B/ is .L1E  ŒE.B/; L1 E .B//-compact [7, Corollary 6.5.10], the expectation EŒX; B is convex .E  ; E/-compact. We summarize some needed results on measurability for convex weak -compact valued Gelfand-integrable mappings in the dual space. A K  -valued mapping X W   E  is a K  -valued random set if X.!/ 2 K  for all ! 2  and if X is scalarly F -measurable. We will show that K  -valued random sets enjoy good measurability properties. Proposition 1. Let X W  ! cwk.Es / be a convex weak -compact valued mapping. The following are equivalent: (a) (b) (c) (d)

X  V 2 F for all m -open subset V of E  . Graph.X / 2 F ˝ B.Es / D F ˝ B.Em  /. X admits a countable dense set of .F ; B.Es //-measurable selections. X is scalarly F -measurable.

Proof. .a/ ) .b/. Recall that any K 2 K  is m -compact and m  w and the Borel tribes B.Es / and B.Em  / are equal. Recall also that Em  is a Lusin metrizable space. By (a), X is an m -compact valued measurable mapping from  into the Lusin metrizable space Em  . Hence Graph.X / 2 F ˝ B.Em  / because Graph.X / D f.!; x  / 2   Em  W dE  .x  ; X.!// D 0g m

and the mapping .!; x  / 7! dE  .x  ; X.!// is F ˝ B.Em  /-measurable. m .b/ ) .a/ is obtained by applying the measurable Projection Theorem (see e.g. [8, Theorem III.23]) and the equality X  V D proj fGraph.X / \ .  V /g: Hence (a) and (b) are equivalent. .b/ ) .c/. Since Es is a Lusin space, by [8, Theorem III-22], X admits a countable dense set of .F ; B.Es //-measurable selections .fn /, that is, X.!/ D w clffn .!/g for all ! 2 . .c/ ) .d /. Indeed one has ı  .x; X.!// D supn hx; fn .!/i for all x 2 E and for all ! 2 , thus proving the required implication. .d / ) .b/. We have already seen that .d / implies that Graph.X / 2 F ˝B.Es /. As B.Es / D B.Em  /, the proof is finished. 

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Corollary 1. Let X W  ! cwk.Es / be a convex weak -compact valued mapping. The following are equivalent: (a) (b) (c) (d)

X  V 2 F for all w -open subset V of E  . Graph.X / 2 F ˝ B.Es /. X admits a countable dense set of .F ; B.Es //-measurable selections. X is scalarly F -measurable.

Proof. .a/ ) .d / is easy. The implications .d / ) .b/, .b/ ) .c/, .c/ ) .d /, .b/ ) .a/ are already known. For further details on these facts, consult Proposition 5.2 and Corollary 5.3 in [5].  Let .Xn /n2N be a sequence of w -closed convex sets, the sequential weak upper limit w -ls Xn of .Xn /n2N is defined by w -ls Xn D fx  2 E  W x  D .E  ; E/- lim xj I xj 2 Xnj g: j !1

Similarly the sequential weak lower limit w -li Xn of .Xn /n2N is defined by w -li Xn D fx  2 E  W x  D .E  ; E/- lim xn I xn 2 Xn g: n!1

The sequence .Xn /n2N weak star (w K for short) converges to a w -closed convex set X1 if the following holds w -ls Xn  X1  w -li Xn

a.s.

Briefly w K- lim Xn D X1 n!1

a.s.

We need the following definition. Definition 1. The Banach space E is weakly compactly generated (WCG) if there exists a weakly compact subset of E whose linear span is dense in E. Every separable Banach space is WCG, and every dual of a separable Banach space (endowed with the dual norm) is WCG. For the sake of completeness we recall the following [11] Theorem 1. Suppose E is WCG (not necessarily separable) and let C and Cn .n D 1; 2; : : :/ be weak -closed, bounded, convex non empty sets of E  . Then ı  .:; Cn / ! ı  .:; C / pointwise on E if and only if the sequence .Cn / is uniformly bounded with w K limit C . Now we provide some applications.

On the Integration of Fuzzy Level Sets

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1 Theorem 2. Let X; Xn.n 2 N/ be a sequence in Lcwk.E  .F / with the following s / property: jX j C jXn j  g for all n 2 N where g is positive integrable. Then the following hold: R R (a)  XdP;  Xn dP , .n 2 N/, are convex weak -compact, (b) If Xn w K converges to X , equivalently, ı  .:; Xn / ! ı  .:; X / pointwise on E, then Z Z  w K- lim Xn dP D XdP: n!1 



1 1 Proof. (a) As jX j  g and jXn j  g for all n 2 N, SGe .X /.F / and SGe .Xn /.F / 1 1 are convex sequentially .LE  ; LE /-compact [7, Corollary 6.5.10], so that s R R   XdP ,  Xn dP , .n D 1; 2; : : :/, are convex weak -compact.

(b) Further by the Strassen theorem [8], we have that ı  .x; 

ı .x;

Z

Z





XdP / D

Xn dP / D

Z

ı  .x; X /dP;

8x 2 E:



Z

ı  .x; Xn /dP;

(1)

8x 2 E:



(2)

Applying Lebesgue’s theorem and (1)–(2) gives lim ı  .x;

n!1

D

Z



Z

Xn dP / D lim



Z

n!1 

ı  .x; Xn /dP

ı  .x; X /dP D ı  .x;

Invoking Theorem 1, we conclude that

R



Z

XdP /: 

Xn dP w K converges to

R



XdP . 

The following concerns the continuous dependence of the Aumann-Gelfand 1 multivalued integral of a cwk.Es /-valued mapping X˛ 2 Lcwk.E  .F / depending s / on a parameter ˛ 20; 1. It has some importance in applications.

Theorem 3. Let X W 0; 1  Es be a convex weakly compact valued mapping satisfying the properties:

(i) jX.!; ˛/j  g.!/; 8!; 2 ; 8˛ 20; 1, where g is a positive integrable function, 1 (ii) For every ˛ 20; 1, X.:; ˛/ 2 Lcwk.E  .F /, s / (iii) For every ! 2 , ˛ 7! X.:; ˛/ from 0; 1 into cwk.Es / is scalarly left continuous. R Then the mapping ˛ 7!  X.:; ˛/dP from 0; 1 into cwk.Es / is scalarly left continuous.

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Proof. Follows the lines of the proof of Theorem 2. By (i) jX˛ j  g for all ˛ 20; 1 with X˛ WD X.:; ˛/. Let ˛n ! ˛. Then X.!; ˛n / scalarly converges to X.!; ˛/,  that is, ı  .x; X.!; R ˛n // ! ı .x; X.!; ˛// for every x 2 E and for every ! 2 . Remember that  X˛ dP is convex weakly compact for every ˛ 20; 1. Further by Strassen’s theorem, we have Z Z  ı  .x; X˛ /dP; 8x 2 E: (3) ı .x; X˛ dP / D 



Applying Lebesgue’s theorem and (3) gives lim ı  .x;

n!1

D

Z



Z

X˛n dP / D lim

n!1 





Z



ı  .x; X˛n /dP

ı .x; X˛ /dP D ı .x;

Z

X˛ dP /: 



3 Random Fuzzy Convex Upper Semicontinuous Integrands Thanks to measurable properties developed in Sect. 2 we present now some applications to a special class of random upper semicontinuous integrands (variables). We recall some definitions that are borrowed from the study of normal integrands (alias random lower semi-continuous integrands) on a general locally convex Suslin space E. A random lower semicontinuous (resp. upper semicontinuous) integrand is a F ˝ B.E/-measurable function X defined on   E such that X.!; :/ is lower semicontinuous (resp. upper semicontinuous). The study of random lower semicontinuous integrands occurs in some problems in Convex Analysis and Variational convergence. See e.g. [28] and the references therein. In the following we will focus on a special class of random upper semicontinuous integrands. Here the terminologies are borrowed from the theory of fuzzy sets initiated by Zadeh [30] and random fuzzy sets initiated by Feron [10] and Puri-Ralescu [25]. According to [30] a fuzzy convex upper semicontinuous variable is a mapping X W E ! Œ0; 1 such that (i) X is upper semicontinuous, (ii) fx 2 E W X.x/ D 1g D 6 ; (iii) X is fuzzy convex, that is, X.x C .1  /y/  min.X.x/; X.y//, for all  2 Œ0; 1 and for all x; y 2 E. A random fuzzy convex upper semicontinuous variable is an F ˝B.E/-measurable mapping X W   E ! Œ0; 1 such that each ! 2 , a mapping X! W E ! Œ0; 1 is

On the Integration of Fuzzy Level Sets

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a fuzzy convex upper semicontinuous variable. By upper semicontinuity and fuzzy convexity, for each ! 2  and for each ˛ 20; 1, the level set X˛ .!/ WD L˛ .X /.!/ WD fx 2 E W X! .x/  ˛g is closed convex. It is clear that the graph of this multifunction belongs to F ˝ B.E/. In particular, this mapping is F -measurable by the measurable projection theorem [8, Theorem III. 23]. Further, thanks to [8, Lemma III.39], for any F ˝ B.E/-measurable mapping ' W   E ! Œ0; C1, the function m.!/ WD supf'.!; x/ W x 2 L˛ .X /.!/g is F -measurable. Similarly the graph of the multifunction fx 2 E W X.!; x/ > 0g WD ŒX! > 0 belongs to F ˝ B.E/. In particular, this mapping is F -measurable by the measurable projection theorem [8, Theorem III. 23]. Further, thanks to [8, Lemma III.39], for any F ˝ B.E/measurable mapping ' W   E ! Œ0; C1, the function ! 7! supf'.!; x/ W x 2 ŒX! > 0g is F -measurable. In particular, if the underlying space E is a separable Banach space, then ŒX! > 0 D fx 2 E W X.!; x/ > 0g is F measurable and so is the mapping ŒX! > 0 so that the mapping ! 7! supfjjxjj W x 2 ŒX! > 0g is F -measurable, further assume that ŒX! > 0 is compact and ! 7! g.!/ WD supfjjxjj W x 2 ŒX! > 0g integrable, then L˛ .X / is convex compact valued and integrably bounded: jL˛ .X /j  g for all ˛ 20; 1, here measurability of g is ensured because of the above measurable properties. Similarly, for each ˛ 2 Œ0; 1Œ, L˛C .X /.!/ WD ŒX! > ˛ is compact valued and F -measurable. The above considerations still hold when the underlying space is the weak star dual Es of a separable Banach space E because Es is a Lusin space, by measurability results developed in Sect. 2. Now we present some convergence properties of the level sets associated with a random fuzzy convex upper semicontinuous integrand. Proposition 2. Let X W E ! Œ0; 1 random fuzzy convex upper semicontinuous integrand with the following properties: (1) fx 2 E W X.!; x/ > 0g is compact, for each ! 2 , (2) g WD jL0C .X /j 2 L1 , (3) The ck.E/-valued mapping ˛ 7! L˛ .X / is scalarly left continuous on 0; 1. Then the following hold R R (a)  L˛ .X /dP D f  f dP W f 2 SL1 ˛ .X / g .˛ 20; 1/ is convex compact. R (b) The ck.E/-valued mapping ˛ 7!  L˛ .X /dP is scalarly left continuous on 0; 1. Proof. With the properties (1)–(2), we see that the level sets L˛ .X / (˛ 20; 1) belong to the space Lck.E/ .; F / of all convex compact valued integrably bounded multifunctions so that the Aumann integral of L˛ .X / Z



L˛ .X /dP D f

Z



f dP W f 2 SL1 ˛ .X / g

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C. Castaing et al.

is convex compact,1 where SL1 ˛ .X / denotes the set of all integrable selections of the convex compact valued multifunction L˛ .X /. We only sketch the proof. See [2, 8] R for details. Indeed, SL1 ˛ .X / is convex weakly compact in L1E so that  L˛ .X /dP is convex weakly compact in E. Making use of Strassen’s formula we have ı  .x  ;

Z



L˛ .X /dP / D

Z

ı  .x  ; L˛ .X //dP; 

8x  2 E  :

Applying Lebesgue’s dominated convergence theorem shows that 





x 7! ı .x ;

Z

L˛ .X /dP / 

is continuous on the closed unit ball equipped with the topology of compact convergence = weak star topology that is compact metrizable with respect to these topologies. From the Banach-Dieudonné theorem we conclude that this mapping is continuous on Ec . Thus .a/ is proved. Taking account of (3), .b/ follows easily.  The following is a dual variant of the preceding result. Proposition 3. Let X W   Es ! Œ0; 1 be a random fuzzy convex upper semicontinuous variable with the following properties: (1) fx  2 Es W X.!; x  / > 0g is weak -compact, for each ! 2 , 1 (2) g WD jLC 0 .X /j 2 L ,  (3) The cwk.Es /-valued mapping ˛ 7! L˛ .X / is scalarly left continuous on 0; 1. Then the following hold R R (a)  L˛ .X /dP D f  f dP W f 2 SL1 ˛ .X / g .˛ 20; 1/ is convex weak -compact. R (b) The cwk.Es /-valued mapping ˛ 7!  L˛ .X /dP is scalarly left continuous on 0; 1. Proof. With the properties (1)–(3), we see that the level sets L˛ .X / (˛ 20; 1) belong to the space Lcwk.E  / .F / of all convex weak -compact valued integrably bounded multifunctions so that the Aumann integral of L˛ .X / Z



L˛ .X /dP D f

Z



f dP W f 2 SL1 ˛ .X / g

is convex weak -compact because the set SL1 ˛ .X / is convex sequentially .L1E  ; L1 E /-compact [7, Corollary 6.5.10]. Thus .a/ is proved, .b/ follows from s Theorem 3. 

R The compactness of  L˛ .X/dP according to Debreu integral is not available here, see also the remarks of Theorem 8 in Hiai-Umegaki [13].

1

On the Integration of Fuzzy Level Sets

11

4 Expectation and Conditional Expectation of Level Sets Now we proceed to the study of the expectation and conditional expectation of the level sets associated with random fuzzy convex upper semicontinuous integrands. The following lemma is crucial for this purpose. Compare with related results by Puri-Ralescu [25] dealing with fuzzy sets on Rd . Lemma 1. Let X be a random fuzzy convex upper semicontinuous integrand X W   Es ! Œ0; 1 with the following properties: (1) fx  2 Es W X.!; x  / > 0g is weak -compact, for each ! 2 , (2) g WD jL0C .X /j 2 L1 , (3) Assume that 0 < ˛1 < ˛2 < : : : < ˛k ! ˛ and that lim ı  .x; L˛k .X /.!// D ı  .x; L˛ .X /.!//

k!1

for all ! 2  and for all x 2 E. Then we have \Z

k1 

L˛k .X /dP D

Z

L˛ .X /dP: 

Proof. Using (3) and applying Strassen’s theorem [8] and Lebesgue’s dominated convergence theorem gives Z

lim ı  .x;

k!1

D

Z

ı  .x; X˛k .!//dP



X˛k .!/dP / D lim



ı  .x; X˛ .!//dP D ı  .x;

Z

k!1 

Z

(4)

X˛ .!/dP /: 

Since the w -topology coincides with the metrizable topology m , by Theorem 15 in [5] we have Z Z \Z   X˛k dP X˛n dP D X˛n dP D m -LS C WD w - ls 



so that C D w K limk!1

R



X˛k .!/dP . Applying Theorem 1 and (4), we have

ı  .x; C / D lim ı  .x; k!1

k1 

Z



X˛k .!/dP / D ı  .x;

Z

X˛ .!/dP / 

12

for all x 2 E, so that

C. Castaing et al.

R

X˛ dP D C by the separability of E, that is, \Z

k1 

X˛k dP D

Z

X˛ dP: 

 The following theorem yields a crucial property of the expectation of the level sets associated with a random fuzzy convex upper semicontinuous integrand. Theorem 4. Let X be a random fuzzy convex upper semicontinuous integrand X W   Es ! Œ0; 1 with the following properties: (1) fx  2 Es W X.!; x  / > 0g is weak -compact, for each ! 2 , (2) g WD jL0C .X /j 2 L1 , (3) For every fixed ! 2 , the cwk.Es /-valued mapping ˛ 7! L˛ .X /.!/ D X˛ .!/ is scalarly left continuous on 0; 1. Then R R the following T hold X˛ dP D k1 X˛k dP whenever 0 < ˛1 < ˛2 < : : : : < ˛k ! ˛.

Proof. Follows from Lemma 1 using the continuity property of the level sets.



Now we proceed to the conditional expectation of the level sets associated with a random fuzzy convex upper semicontinuous integrand defined on the dual space Es . For this purpose we need to recall and summarize the existence and uniqueness 1 of the conditional expectation in Lcwk.E  .F / [4, 13, 29]. In particular, existence s / results for conditional expectation in Gelfand integration can be derived from the multivalued Dunford-Pettis representation theorem, see [4]. A fairly general version of conditional expectation for closed convex integrable random sets in the dual of a separable Fréchet space is obtained by Valadier [29, Theorem 3]. Here we need only a special version of this result in the dual space Es . Theorem 5. Let € be a closed convex valued integrable random set in Es . Let B be a sub--algebra of F . Then there exists a closed convex B-measurable mapping † in Es such that: (1) † is the smallest closed convex B-measurable mapping ‚ such that 8u 2 S€1 , E B u.!/ 2 ‚.!/ a.s. (2) † is the unique closed convex B-measurable mapping ‚ such that 8v 2 L1 R .B/, Z





ı .v; €/dP D

Z

ı  .v; ‚/dP:



(3) † is the unique closed convex B-measurable mapping such that S†1 D cl .E B .S€1 // where cl denotes the closure with respect to .L1E  .B/; L1 E .B//.

On the Integration of Fuzzy Level Sets

13

Theorem 5 allows to obtain the weak compactness of the conditional expectation of convex weakly compact valued integrably bounded mappings in E with strong separable dual. Indeed if F WD Eb is separable and if € is a convex weakly compact valued measurable mapping in E with €.!/  ˛.!/B E where ˛ 2 L1R , then applying Theorem 4 to F  gives †.!/ D E B €.!/  E  with †.!/  E B ˛.!/B E  where B E  is the closed unit ball in E  . As S€1 is .L1E ; L1 E  /compact, S†1 D E B .S€1 /  L1E . Whence †.!/  E a.s. See [29, Remark 4, page 10] for details. The following existence theorem of conditional expectation for convex weak compact valued Gelfand-integrable mappings follows from a multivalued version of the Dunford-Pettis theorem in the dual space [4, Theorem 7.3]. In particular, it provides the weak -compactness of conditional expectation for integrably bounded weak -compact valued scalarly measurable mappings with some specific properties. 1 Theorem 6. Given € 2 Lcwk.E  .F / and a sub--algebra B of F , there s / 1 exists a unique (for equality a.s.) mapping † WD E B € 2 Lcwk.E  .B/, that is s / the conditional expectation of € with respect to B, which enjoys the following properties: R R (a)  ı  .v; †/dP D  ı  .v; €/dP for all v 2 L1 E .B/. (b) †  E B j€jB E  a.s. 1 (c) S†1 .B/ is .L1E  ŒE.B/; L1 E .B//- compact (here S† .B/ denotes the set of 1 all LE  ŒE.B/ selections of †) and satisfies

ı  .v; E B S€1 .F // D ı  .v; S†1 .B// for all v 2 L1 E .B/. (d) E B is increasing: €1  €2 a.s. implies E B €1  E B €2 a.s. Now we need at first a conditional expectation version for Lemma 1. Lemma 2. Let B be a sub--algebra of F and let X be a random fuzzy convex upper semicontinuous integrand X W   Es ! Œ0; 1 with the following properties: (1) fx  2 Es W X.!; x  / > 0g is weak -compact, for each ! 2 , (2) g WD jL0C .X /j 2 L1 , (3) Assume that 0 < ˛1 < ˛2 < : : : : < ˛k ! ˛ and that lim ı  .x; L˛k .X /.!// D ı  .x; L˛ .X /.!//

k!1

for all ! 2  and for all x 2 E.

Let E B X˛ be the conditional expectation of the level sets L˛ .X / WD X˛ , ˛ 20; 1. Then we have \ E B X˛k D E B X˛ : k1

14

C. Castaing et al.

Proof. We follow some lines of the proof of Lemma 1. But here we need a careful look using some convergence results in conditional expectation. By (1)–(3) we have L˛ .X /.!/ WD X˛ .!/  Y .!/ for all ˛ 20; 1 and for all ! 2  where Y is a weak -compact valued integrably bounded multifunction. By Theorem 6 and by our assumption (1)–(3) the conditional expectation E B X˛ is convex weak -compact valued, B-measurable and integrably bounded with E B X˛ .!/  .E B g/.!/ B E  for all ˛ 20; 1 and for all ! 2 . Since the weak -topology coincides with the metrizable topology m , by Theorem 5.4 in [5] the multifunction V .!/ D w - ls E B X˛n .!/ D m -LS E B X˛n .!/ D

\

E B X˛k .!/

k1

is w -compact and B-measurable so that V .!/ D w K lim E B X˛k .!/: k!1

By Theorem 1, we have lim ı  .x; E B X˛k .!// D ı  .x; V .!//

k!1

for all ! 2  and for all x 2 E. By the dominated convergence theorem for conditional expectations, we have ı  .x; V .!// D lim ı  .x; E B X˛k .!// D lim E B ı  .x; X˛k .!// k!1

k!1

D E B ı  .x; X˛ .!// D ı  .x; E B X˛ .!// so that E B X˛ D V by the separability of E, noting that E B X˛ .!/ is convex weak -compact, and that E B X˛ .!/ D

\

E B X˛k .!/

k1

for all ! 2 . Now we proceed to the conditional expectation of the level sets associated with such a random fuzzy convex upper semicontinuous integrand X W   Es ! Œ0; 1. Theorem 7. Let B be a sub--algebra of F and let X be a random fuzzy convex upper semicontinuous integrand X W   Es ! Œ0; 1 with the following properties: (1) fx  2 Es W X.!; x  / > 0g is weak*-compact, for each ! 2 , (2) g WD jL0C .X /j 2 L1 , (3) For every fixed ! 2 , the cwk.Es /-valued mapping ˛ 7! L˛ .X /.!/ D X˛ .!/ is scalarly left continuous on 0; 1.

On the Integration of Fuzzy Level Sets

15

Then the following Thold E B X˛ .!/ D k1 E B X˛k .!/ for every ! 2  and every ˛ 20; 1, whenever ˛1 < ˛2 : : : < ˛k ! ˛. Proof. Here we will use again the monotonicity of the conditional expectation and the monotonicity of the level sets, namely for ˛  ˇ; X ˇ  X˛ and E B X ˇ  E B X˛ . We have to check that \ E B X˛k .!/ ( ) E B X˛ .!/ D k1

for every ! 2  whenever 0 < ˛1 < ˛2 < : : : < ˛k ! ˛. By the continuity of the level sets (3) and the dominated convergence theorem for the conditional expectation, we have lim ı  .x; E B X˛k .!// D lim E B ı  .x; X˛k .!//

k!1

k!1

D E B ı  .x; X˛ .!// D ı  .x; E B X˛ .!// for all x 2 E, so that the desired inclusion follows from the arguments developed  in first part of the proof of Lemma 2. With the above considerations, we produce a general result on the conditional 1 expectation of a convex weak -compact valued X˛ 2 Lcwk.E  .F / depending on s / the parameter ˛ 20; 1.

Theorem 8. Let B be a sub--algebra of F , and let X W 0; 1  Es be a convex weak -compact valued mapping with the following properties: (1) jX.!; ˛/j  g 2 L1 for all .!; ˛/ 2 0; 1, (2) For each ! 2 , X.!; :/ is scalarly left continuous on 0; 1, (3) For every fixed ˛ 20; 1, X:.; ˛/ is scalarly F -measurable.

Then the convex weak -compact valued conditional expectation E B X˛ of the mapping X˛ enjoys the properties (a) For each ! 2 , ˛ 7! E B X˛ .!/ is scalarly left continuous on 0; 1, (b) For each ˛ 20; 1, ! 7! E B X˛ .!/ is scalarly B-measurable on , (c) Assume further that ˛ 7! X.!; ˛/ is decreasing, for every fixed !, i.e. ˛ < ˇ 2 0; 1 implies X.!; T ˇ/  X.!; ˛/, then 0 < ˛1 < ˛2 < : : : < ˛k ! ˛ implies E B X˛ .!/ D k1 E B X˛k .!/: Similarly we have

Theorem 9. Let X W 0; 1  Es be a convex weak -compact valued mapping with the following properties: (1) jX.!; ˛/j  g 2 L1 for all .!; ˛/ 2 0; 1, (2) For each ! 2 , X.!; :/ is scalarly left continuous on 0; 1, (3) For every fixed ˛ 20; 1, X.:; ˛/ is scalarly F -measurable on .

16

C. Castaing et al.

Then the convex weak -compact valued mapping EX˛ WD the properties

R



X.!; ˛/dP enjoys

(a) ˛ 7! EX˛ is scalarly left continuous on 0; 1, (b) Assume further that ˛ 7! X.!; ˛/ is decreasing, for every fixed !, then 0 < T ˛1 < ˛2 < : : : < ˛k ! ˛ implies EX˛ D k1 EX˛k : Corollary 2. Assume that E D Rd and that .; F ; P / has no atoms. Let X W 0; 1  Rd be a compact valued mapping with the following properties:

(1) jX.!; ˛/j  g 2 L1 for all .!; ˛/ 2 0; 1, (2) For each ! 2 , X.!; :/ is scalarly left continuous on 0; 1, (3) For every fixed ˛ 20; 1, X.:; ˛/ is scalarly F -measurable on . R Then the convex compact valued mapping EX˛ WD  X.!; ˛/dP enjoys the properties (a) ˛ 7! EX˛ is scalarly left continuous on 0; 1, (b) Assume further that ˛ 7! X.!; ˛/ is decreasing, for every fixed !, then 0 < T ˛1 < ˛2 < : : : < ˛k ! ˛ implies EX˛ D k1 EX˛k : Proof. Since E D Rd and .; F ; P / has no atoms, by [8, Theorem IV-17], EX˛ is convex compact in Rd . 

Using the notations of the preceding results and the Negoita-Ralescu representation theorem [25, Lemma 1] or Höhle and Šostak [14, Lemma 6.5.1], we mention a useful result. Lemma 3. Let .C˛ /˛2Œ0;1 be a family of convex weak -compact subsets in E  with the properties: (1) (2) (3) (4)

C0 WD Es , Cˇ  C˛ ; 8˛ < ˇ 2 Œ0; 1, C˛  rB E  for all ˛ 20; 1, ˛ 7! ı  .x; C˛ / is left continuous on 0; 1 for all x 2 E.

There is a unique fuzzy convex upper semicontinuous variable ' W Es ! Œ0; 1 such that fx  2 Es W '.x  /  ˛g D C˛ , for every ˛ 20; 1, where ' is given by '.x  / D supf˛ 2 Œ0; 1 W x  2 C˛ g: Proof. Step 1. We use at first some arguments and results developed in Lemma 1. Let 0 < ˛1 < ˛2 ; : : : < ˛k ! ˛. By (4) we have lim ı  .x; C˛k / D ı  .x; C˛ /

k!1

(5)

On the Integration of Fuzzy Level Sets

17

for all x 2 E. Note that by (2) and (3), .C˛n / is uniformly bounded and decreasing. Since the w -topology coincides with the metrizable topology m on bounded subsets in the weak dual, by Theorem 5.4 in [5] C WD w - ls C˛n D m -LS C˛n D

\

C˛k

k1

so that C D w K limk!1 C˛k . Applying Theorem 1, we have ı  .x; C / D lim ı  .x; C˛k / k!1

(6)

for all x 2 E, so that using (5) and (6) we have that ı  .x; C˛ / D ı  .x; C / for all x 2 E, whence C˛ D C by the separability of E, that is, \

k1

C˛k D C˛ :

(7)

Step 2. fx  2 Es W '.x  /  ˛g D C˛ , for every ˛ 20; 1. Here we may apply the arguments given in the Negoita-Ralescu representation theorem and the results in Step 1 to get this equality. Let x  2 C˛0 . Then ˛0 2 f˛ 2 Œ0; 1 W x  2 C˛ g which implies '.x  / D supf˛ 2 Œ0; 1 W x  2 C˛ g  ˛0 thus x  2 Œ'  ˛0 . To show the converse inclusion, let x  2 Œ'  ˛0 . Then '.x  / D supf˛ 2 Œ0; 1 W x  2 C˛ g  ˛0 : If '.x  / > ˛0 , there exists ˛1  ˛0 with x  2 C˛1 . But then we have C˛1  C˛0 by (2), thus x  2 C˛0 . Assume that '.x  / D ˛0 . Then there exists .˛k / such that x  2 C˛k for each k and that ˛k " ˛0 . But then x 2 by Step 1.

1 \

nD1

C˛k D C˛0

18

C. Castaing et al.

Applying Theorem 4 and Lemma 3, we get the expectation of random fuzzy convex integrands. Theorem 10. Let B be a sub--algebra of F and let X be a random fuzzy convex upper semicontinuous integrand X W   Es ! Œ0; 1 with the following properties: (1) fx  2 Es W X.!; x  / > 0g is weak -compact, for each ! 2 , (2) g WD jL0C .X /j 2 L1 , (3) For every fixed ! 2 , the K  -valued mapping ˛ 7! L˛ .X /.!/ D X˛ .!/ is scalarly left continuous on 0; 1. Q / on Es Then there exist a unique fuzzy convex upper semicontinuous function E.X satisfying Q /˛ D ŒE.X

Z



X˛ dP; 8˛ 20; 1I

Q / is the fuzzy expectation of X . E.X R R Proof. Apply Lemma 3 to the family .C˛ D  X˛ dP /˛2Œ0;1 with C0 WD  X0 dP D E  by taking account of Proposition 3 and Theorem 4.  Now it is possible to provide the fuzzy conditional expectation of random fuzzy convex upper semicontinuous integrand. Theorem 11. Let B be a sub--algebra of F and let X be a random fuzzy convex upper semicontinuous integrand X W   Es ! Œ0; 1 with the following properties: (1) fx  2 Es W X.!; x  / > 0g is weak -compact, for each ! 2 , (2) g WD jL0C .X /j 2 L1 , (3) For every fixed ! 2 , the K  -valued mapping ˛ 7! L˛ .X /.!/ D X˛ .!/ is scalarly left continuous on 0; 1. Let E B X˛ D E.X˛ jB/ be the conditional expectation of X˛ for every ˛ 2 Œ0; 1 and let '! .x  / WD supf˛ 2 Œ0; 1 W x  2 E B X˛ .!/g for every ! 2  and for every ˛ 2 Œ0; 1. Then the following hold fx  2 Es W '! .x  /  ˛g D E B X˛ .!/ Q jB/ WD ' is the fuzzy conditional for every ! 2  and for every ˛ 20; 1; E.X expectation of X .

On the Integration of Fuzzy Level Sets

Proof. Step 1. Theorem 7

19

1 By Theorem 6, recall that E B X˛ 2 Lcwk.E  / .B/ and by virtue of s

E B X˛ .!/ D

\

E B X˛k .!/

k1

for every ! 2  and every ˛ 20; 1, whenever ˛1 < ˛2 ; : : : < ˛k ! ˛. Step 2. Let ˛0 20; 1 and let ! 2 . Let x  2 E B X ˛0 .!/. Then ˛0 2 f˛ 2 Œ0; 1 W x  2 E B X˛ .!/g which implies '! .x  / D supf˛ 2 Œ0; 1 W x  2 E B X ˛ .!/g  ˛0 thus x  2 Œ'!  ˛0 . To show the converse inclusion, let x  2 Œ'!  ˛0 . Then '! .x  / D supf˛ 2 Œ0; 1 W x  2 E B X˛ .!/g  ˛0 : If '! .x  / > ˛0 , there exists ˛1  ˛0 with x  2 E B X˛1 .!/. But then we have E B X˛1 .!/  E B X ˛0 .!/ by the monotonicity of the conditional expectation, thus x  2 E B X˛0 .!/. Assume that '! .x  / D ˛0 . Then there exists .˛k / such that x  2 E B X˛k .!/ for each k and that ˛k " ˛0 . But then x 2

1 \

kD1

E B X˛k .!/ D E B X˛0 .!/

by using Step 1.



It is worth to state the relationships between the fuzzy expectation and the conditional expectation of a random upper semicontinuous fuzzy convex integrand X . When E is a reflexive separable Banach space and X 2 LE1 .; F ; P / and B is a sub--algebra of F , EX is the expectationRof X and E B XR the conditional expectation of X , then, for any B 2 B, we have B E B XdP D B XdP , now we have a similar equality if we deal with a random upper semicontinuous fuzzy convex Q / and fuzzy conditional expectation E.X Q jB/. integrand X , fuzzy expectation E.X Namely the following equality holds Z

B

Q jB/˛ dP D ŒE.X

Z

B

B

E X˛ dP D

for every B 2 B and for every ˛ 20; 1.

Z

B

Q X˛ dP D ŒE.X1 B /˛

20

C. Castaing et al.

Our results can be applied to the convergence of convex weakly compact valued level sets of a random upper semicontinuous integrand defined on a separable reflexive Banach space using the fuzzy expectation and the fuzzy conditional expectation. Next we will provide some SLLN results for fuzzy random variables in a separable Banach space.

5 SLLN for Fuzzy Random Variables in a Separable Banach Space Let c.E/ (resp. k.E/) (resp. cwk.E/) (resp. ck.E/) denote the set of all nonempty closed (resp. compact) (resp. convex weakly compact) (resp. convex compact) subsets in E. Here we focus on convergence in the Polish space .ck.E/; dH / where dH is the Hausdorff distance on ck.E/. Let us recall and summarize some needed results. Lemma 4. Let .Xn / be a sequence in k.E/. If n

lim dH .

n!1

1X coXi ; X / D 0 n i D1

for some X 2 ck.E/, then n

lim dH .

n!1

1X Xi ; X / D 0 n i D1

Proof. See e.g. Arstein-Hansen [1], de Blasi and Tomassini [9], Hiai [12]. The following result is borrowed from Castaing and Raynaud de Fitte [6, Theorem 4.8]. Theorem 12. Let .Xn / be a pairwise independent identically distributed sequence of integrably bounded ck.E/-valued such that g WD supn2N jXn j  r is integrable, then EŒXn  D EŒX1  2 ck.E/; 8n 2 N and n

lim dH .

n!1

1X Xi ; EŒX1 / D 0 a.s. n i D1

On the Integration of Fuzzy Level Sets

21

Remarks. It is important to have the convexity and the norm compactness [2] of EŒXn  D EŒX1  because EŒX1  is the norm compactness limit in our SLLN. It is also worth to note that if .Xn / is a sequence of convex compact valued integrably bounded i.i.d. random sets, the random variable g D supn jXn j is necessarily constant (with finite value). See [6] for details. Now we provide some applications to the SLLN for pairwise i.i.d. compact valued integrably bounded random sets. Theorem 13. Let .Xn / be a pairwise independent identically distributed sequence of integrably bounded k.E/-valued random sets in E such that g WD supn2N jXn j  ˛ is integrable. Then EŒcoXn  D EŒcoX1  2 ck.E/; 8n 2 N and n

lim dH .

n!1

1X Xi ; EŒcoX1 / D 0 a.s. n i D1

Proof. Since .coXn / is pairwise independent identically distributed integrably bounded ck.E/-valued, Theorem 12 shows that n

1X coXi ; EŒcoX1 / D 0 dH . n i D1

a.s.

Invoking Lemma 4 yields n

lim dH .

n!1

1X Xi ; EŒcoX1 / D 0 n i D1

a.s. 

The following is an important consequence. Theorem 14. Assume E D Rd and .; F ; P / has no atom. Let .Xn / be a pairwise independent identically distributed sequence of integrably bounded k.E/-valued random sets in E such that g WD supn2N jXn j  ˛ is integrable. Then EŒXn  D EŒX1  2 ck.Rd /; 8n 2 N and n

lim dH .

n!1

1X Xi ; EŒX1 / D 0 a.s. n i D1

22

C. Castaing et al.

Proof. Since .coXn / is pairwise independent identically distributed integrably bounded ck.E/-valued, Theorem 12 shows that n

dH .

1X coXi ; EŒcoX1 / D 0 a.s. n i D1

By invoking Lemma 4 it follows that n

lim dH .

n!1

1X Xi ; EŒcoX1 / D 0 n i D1

a.s.

But E D Rd and .; F ; P / has no atom, thus EŒXn  D EŒcoXn  D EŒcoX1  D EŒX1  2 ck.Rd /; 8n 2 N using Ljapunov’s Theorem [8, Theorem IV.17].



d

Remark. If E D R , .; F ; P / has no atom and X is compact valued integrably bounded, then EŒX  is compact convex, this result is not valid in a separable Banach space. Now is a version of SLLN in the primal space for fuzzy random variables. Theorem 15. Let .X n /n2N be a sequence of random fuzzy convex upper semicontinuous variable X n W   E ! Œ0; 1 with the following properties: (1) fx 2 E W X n .!; x/ > 0g is compact, for each n 2 N and for each ! 2 , (2) g WD supn jL0C .X n /j is integrable. Assume that (3) .X˛n D L˛ .X n //n2N is pairwise i.i.d, for each ˛ 20; 1, (4) .X˛nC D L˛C .X n //n2N is pairwise i.i.d, for each ˛ 2 Œ0; 1Œ. Then we have, for every ˛ 20; 1, n

1X i X ; EŒX˛1 / D 0 a.s. lim dH . n!1 n i D1 ˛ n

lim dH .

n!1

1X i 1 X ; EŒcoX˛C / D 0 a.s. n i D1 ˛C

Assume further that the following condition is satisfied: (5) Given " > 0, there exists a partition 0 D ˛0 < ˛1 < : : : < ˛m D 1 of Œ0; 1 such that max1km dH .EŒcoX 1C ; EŒX˛1k / < ". ˛k1

Then we have

On the Integration of Fuzzy Level Sets

23 n

lim sup dH .

n!1 ˛20;1

1X i X ; EŒX˛1 / D 0: n i D1 ˛

n Proof. (a) Since .X˛n /n2N and .X˛C /n2N are independent identically distributed compact valued random variables, we have, for every ˛ 20; 1, by Theorem 13, n

1X i X ; EŒX˛1 / D 0 lim dH . n!1 n i D1 ˛

a.s.

(8)

a.s.

(9)

n

lim dH .

n!1

1X i 1 X ; EŒcoX˛C / D 0 n i D1 ˛C

(b) Let " > 0 be given. There exists a partition 0 D ˛0 < ˛1 < : : : < ˛m D 1 of Œ0; 1 such that max dH .EŒcoX 1C ; EŒX˛1k / < ": ˛k1

1km

Let ˛ 20; 1. Then there exist k (depending on ˛) such that ˛k1 < ˛  ˛k . We will use some elementary facts: n

n

n

1X i 1X i 1X i X˛k  X˛  X C : n i D1 n i D1 n i D1 ˛k1 EŒX˛1k   EŒX˛1   EŒcoX 1C : ˛k1

Now we have the estimation n

dH .

n

1X i 1X i X˛ ; EŒX˛1 /  dH . X C ; EŒX˛1k / n i D1 n i D1 ˛k1 n

CdH .

n

1X i 1X i X˛k ; EŒcoX 1C /  dH . X ; EŒX˛1k / ˛k1 n i D1 n i D1 ˛k n

CdH .

1X i X C ; EŒcoX 1C / C 2dH .EŒcoX 1C ; EŒX˛1k / ˛k1 ˛k1 n i D1 ˛k1 n

 max dH . 1km

n

1X i 1X i X˛k ; EŒX˛1k / C max dH . X C ; EŒcoX 1C / ˛k1 1km n i D1 n i D1 ˛k1

C2 max dH .EŒcoX 1C ; EŒX˛1k / WD I1 C I2 C I3 : 1km

˛k1

24

C. Castaing et al.

From (8) it follows that n

lim dH .

n!1

1X i X ; EŒX˛1k / D 0 a.s. n i D1 ˛k

and from (9), we have that n

lim dH .

n!1

1X i X C ; EŒcoX 1C / D 0 a.s. ˛k1 n i D1 ˛k1

for every k D 1; : : : ; m. Hence for a.s. ! 2 , we have n

max dH .

1km

1X i X .!/; EŒX˛1k / ! 0; as n ! 1: n i D1 ˛k

that is, I1 ! 0; n ! 1

(10)

Similarly, for a.s. ! 2 , we have I2 ! 0; n ! 1:

(11)

We have I3 D 2 max dH .EŒcoX 1C ; EŒX˛1k /  2": ˛k1

1km

Finally we have n

dH .

1X i X .!/; EŒX˛1 / < I1 C I2 C 2" n i D1 ˛

(12)

for all ˛ 20; 1: Since I1 , I2 and " do not depend on ˛, then from (12), for a.s. ! 2  we get n

1X i sup dH . X .!/; EŒX˛1 / < I1 C I2 C 2": n i D1 ˛ ˛20;1 Passing to the limit when n goes to 1 in the preceding inequality yields n

lim dH .

n!1

1X i X .!/; EŒX˛1 /  2" a.s. n i D1 ˛

On the Integration of Fuzzy Level Sets

25

Whence n

lim sup dH .

n!1 ˛20;1

1X i X .!/; EŒX˛1 / D 0 n i D1 ˛

since " is arbitrary. Here is an important variant. Theorem 16. Assume E D Rd and .; F ; P / has no atom. Let .X n /n2N be a sequence of random fuzzy convex upper semicontinuous variable X n W   E ! Œ0; 1 with the following properties: (1) fx 2 E W X n .!; x/ > 0g is compact, for each n 2 N and for each ! 2 , (2) g WD supn jL0C .X n /j is integrable. Assume that (3) .X˛n D L˛ .X n //n2N is pairwise i.i.d, for every ˛ 20; 1, (4) .X˛nC D L˛C .X n //n2N is pairwise i.i.d, for every ˛ 2 Œ0; 1Œ. Then we have n

lim sup dH .

n!1 ˛20;1

1X i X ; EŒX˛1 / D 0 n i D1 ˛

a.s.

n Proof. (a) Since .X˛n /n2N and .X˛C /n2N are pairwise independent, identically distributed compact valued random variables, by Theorem 14, we have n

lim dH .

n!1

1X i X ; EŒX˛1 / D 0 n i D1 ˛

a.s.

(13)

for every ˛ 20; 1 n

lim dH .

n!1

1X i 1 / D 0 X ; EŒX˛C n i D1 ˛C

a.s.

(14)

for every ˛ 2 Œ0; 1Œ. (b) Let " > 0 be given. Using a technique similar to the one developed in Joo et al [17, Theorem 3.1], we provide a partition 0 D ˛0 < ˛1 < : : : < ˛m D 1 of Œ0; 1 such that max dH .EŒX 1C ; EŒX˛1k / < ":

1km

˛k1

26

C. Castaing et al.

Let ˛ 20; 1. Then there exist k (depending on ˛) such that ˛k1 < ˛  ˛k . We will use some elementary facts: n

n

n

1X i 1X i 1X i X˛k  X˛  X C : n i D1 n i D1 n i D1 ˛k1 EŒX˛1k   EŒX˛1   EŒX 1C : ˛k1

Now we have the estimation n

dH .

n

n

 dH .

n

1X i 1X i 1X i X˛ ; EŒX˛1 /  dH . X C ; EŒX˛1k /CdH . X ; EŒX 1C / ˛k1 n i D1 n i D1 ˛k1 n i D1 ˛k n

1X i 1X i X˛k ; EŒX˛1k /CdH . X C ; EŒX 1C /C2dH .EŒX 1C ; EŒX˛1k / ˛k1 ˛k1 n i D1 n i D1 ˛k1 n

 max dH . 1km

n

1X i 1X i X˛k ; EŒX˛1k / C max dH . X C ; EŒX 1C / ˛k1 1km n i D1 n i D1 ˛k1

C2 max dH .EŒX 1C ; EŒX˛1k / WD I1 C I2 C I3 : 1km

˛k1

Now the rest of the proof is identical to the last part of the proof of Theorem 15 by noting that I3 is  2" and I1 ! 0 and I2 ! 0 when n ! 1. 

6 Fuzzy Martingale and Integrand Martingale We discuss in this section the concept of fuzzy martingale and integrand martingale and provide some related convergence results. Let .Fn /n2N be an increasing sequence of sub -algebras of F such that F is the -algebra generated by [n2N Fn . Taking into account the results and notations developed in Sect. 4, the expected value (or expectation) of the fuzzy convex upper semicontinuous variable Q n / such that the level set ŒE.X Q n /˛ D X˛n X n can be defined as a fuzzy variable E.X nC1 for every ˛ 20; 1 and also the fuzzy conditional expectation of X with respect to Fn can be defined as an Fn ˝ B.E/-measurable, upper semicontinuous fuzzy Q nC1 jFn / such that ŒE.X Q nC1 jFn /˛ D E Fn X˛nC1 for every convex integrand E.X n n Q nC1 jFn / for all n 2 N. Using the ˛ 20; 1. .X / is a fuzzy martingale if X D E.X n Fn nC1 level sets, it turns out that X˛ D E X˛ for each ˛ 20; 1 so that the adapted sequence .X˛n /n2N is a convex weakly compact valued L1 -bounded martingale in 1 Lcwk.E/ depending on the parameter ˛ 2 Œ0; 1. It is clear that one may consider the notion of fuzzy submartingale or fuzzy supermartingale by replacing the above Q nC1 jFn / or X n  E.X Q nC1 jFn /. Puri and Ralescu [26] treat equality by X n  E.X the convergence of a fuzzy submartingale .X n / by using a different notion of fuzzy

On the Integration of Fuzzy Level Sets

27

conditional expectation and by assuming that X n takes its values in a subspace of fuzzy sets u 2 FcL with the property that the function ˛ 7! L˛ .u/ is Lipschitz with respect to the Hausdorff distance dH .L˛ .u/; Lˇ .u//  C j˛  ˇj, for every ˛; ˇ 20; 1, where C is a positive constant. The above considerations lead to martingales depending on a parameter and integrand martingales independently of the structure of fuzzy sets. Now we provide a result of existence of conditional expectation for normal integrands on a separable Banach space E. A mapping ‰ W   E ! R is a F - normal integrand if it satisfies (a) ‰.!; :/ N is lower semicontinuous on E for all ! 2 , (b) ‰ is F B.E/-measurable.

Let us recall a result of existence of conditional expectation for this class of normal integrands [3, Theorem 5.2] and [16]. Theorem 17. Let ‰ W   E ! R be a F -normal integrand satisfying

(i) There exist a 2 L1RC .; F ; P /, b 2 RC and u0 2 L1E .; F ; P / such that a.!/  bjjx  u0 .!/jjE  ‰.!; x/

8.!; x/ 2   E:

(ii) ‰.:; :u.:// is integrable for all u 2 L1E .; F ; P /.

Let G be a sub--algebra of F .Then there exists a G -normal integrand E G ‰ W   E ! R such that Z Z E G ‰.!; u.!//dP .!/ D ‰.!; u.!//dP .!/ A

A

for all u 2 L1E .; G ; P / and for all A 2 G . Further, the integrand E G ‰ is unique modulo the sets of the form N  E, where N is a P -negligible set in G . E G ‰ is the conditional expectation of ‰ relative to G . Using the conditional expectation of normal integrands, we may define the notion of lower semicontinuous integrand martingale as follows. Definition 2. Let .Fn /n2N be an increasing sequence of sub -algebras of F such that F is the -algebra generated by [n2N Fn and ‰n W   E ! RC .n 2 N/ be a Fn -normal integrand. The sequence .‰n ; Fn /n2N of Fn -normal integrands is a lower semicontinuous integrand martingale if Z

A

‰n .!; u.!//dP D

Z

E Fn ‰nC1 .!; u.!//dP A

for all A 2 Fn , for all u 2 L1E .; Fn ; P / and for all n 2 N: Now we provide an epiconvergence result for integrand martingales.

28

C. Castaing et al.

Theorem 18. Let .Fn /n2N be an increasing sequence of sub -algebras of F such that F is the -algebra generated by [n2N Fn , ‰ W   E ! RC a F -normal integrand such that ‰.:; u.:// is integrable for all u 2 L1E .; F ; P /. Let E Fn ‰ .n 2 N/ be the conditional expectation of ‰ relative to Fn whose existence is given by Theorem 17. Then, for each u 2 L1E .; F ; P / the following variational inequality holds: sup lim sup inf ŒE Fn ‰.!; y/ C kjju.!/  yjjE   ‰.!; u.!// a.s. k2N n!1 y2E

Proof. For each k 2 N, set ‰ k .!; x/ D inf Œ‰.!; y/ C kjjx  yjjE  y2E

8.!; x/ 2   E:

Then we note that 0  ‰ k .!; x/  ‰ kC1 .!; x/  ‰.!; x/ sup ‰ k .!; x/ D ‰.!; x/ k2N

8k 2 N

8.!; x/ 2   E:

8.!; x/ 2   E:

Let u 2 L1E .; F ; P /. Let p 2 N. Since E is separable, applying the measurable selection theorem [8, Theorem III-22], it is not difficult to provide an F -measurable mapping vk;p;u W  ! E such that 0  ‰.!; vk;p;u .!// C kjju.!/  vk;p;u .!/jjE  ‰ k .!; u.!// C

1 p

for all ! 2 , so that ! 7! ‰.!; vk;p;u .!// and ! 7! kjju.!/  vk;p;u .!/jjE are integrable, and so vk;p;u 2 L1E .; F ; P /. By our assumption, ! 7! ‰.!; vk;p;u .!// is integrable, too. Applying Lévy’s theorem yields a negligible set Nk;p;u such that, for all ! … Nk;p;u , lim E Fn Œ‰.!; vk;p;u .!// D ‰.!; vk;p;u .!//:

n!1

Then we deduce lim sup inf ŒE Fn ‰.!; y/ C kjju.!/  yjjE  n!1 y2E

 lim supŒE Fn ‰.!; vk;p;u .!// C kjju.!/  vk;p;u .!/jjE  n!1

D‰.!; vk;p;u .!// C kjju.!/  vk;p;u .!/jjE ‰ k .!; u.!// C

1 p

8! … Nk;p;u :

On the Integration of Fuzzy Level Sets

29

Set Nu WD [k2N;p2N Nk;p;u . Then Nu is negligible. Taking the supremum on k 2 N in the extreme terms yields sup lim sup inf ŒE Fn ‰.!; y/ C kjju.!/  yjjE 

k2N n!1 y2E

 sup ‰ k .!; u.!// D ‰.!; u.!// k2N

8! … Nu : 

Theorem 19. Let Xn W 0; 1 ! RC with the properties (a) Xn .:; ˛/ is Fn -measurable for all n 2 N, for all ˛ 2 Œ0; 1, (b) jXn .!; ˛/  Xn .!; ˇ/j  C.!/j˛  ˇj, for all n 2 N, for all ! 2 , for all ˛; ˇ 2 Œ0; 1, where C is a positive integrable function, (c) 0  Xn .!; ˛/  €.!/, for all n 2 N, for all ! 2 , where € is a positive integrable function, (d) For each ˛ 2 Œ0; 1, .Xn˛ / D .Xn .:; ˛// is a martingale. Then there exists an L12C C€ -bounded RC -valued C -Lipschitz integrand X1 W   Œ0; 1 ! R satisfying ˛ jD0 lim sup jXn˛  E Fn X1

n!1 ˛2Œ0;1

a.s.

Proof. Since .Xn˛ / is an L1 -bounded martingale for each ˛ 2 Œ0; 1, there exists ˛ Y1 2 L1R .F / such that ˛ .!/ lim Xn˛ .!/ D Y1

n!1

a.s.

Passing to the limit when n goes to 1 in the inequality jXn˛ .!/  Xnˇ .!/j  C.!/j˛  ˇj we get ˛ ˇ .!/  Y1 .!/j  C.!/j˛  ˇj jY1

for a.s. ! 2  and for all ˛; ˇ 2 Q, where Q WD .˛m /m2N is a dense sequence in Œ0; 1, so that there is a negligible set N such that ˇ ˛ .!/j  C.!/j˛  ˇjj .!/  Y1 jY1

8! 2  n N

30

C. Castaing et al.

˛ for all ˛; ˇ 2 Q. Let us set for all .!; ˛/ 2   Q, Z1 .!; ˛/ D Y1 .!; ˛/ for ! 2  n N and Z1 .!; ˛/ D 0 for ! 2 N . Then Z1 .!; ˛/ is a C -Lipschitzean integrand on   Q. Let us set

X1 .!; r/ D inf Œ C.!/jr  ˛j C Z1 .!; ˛/ ˛2Q

for all .!; r/ 2   Œ0; 1. Then we have jX1 .!; r/  X1 .!; s/j  C.!/jr  sj; 8.!; r; s/ 2   Œ0; 1  Œ0; 1 and X1 .:; r/ is measurable for any fixed r 2 Œ0; 1 thanks to the measurable projection theorem [8, Theorem III. 23]. It is not difficult to check that X1 is an L12C C€ -bounded RC -valued C -Lipschitz integrand, because 0  X1 .!; r/  C.!/jr ˛jCZ1 .!; ˛/  2C.!/CZ1 .!; ˛/  2C.!/C€.!/ for all !; r 2   Œ0; 1. Now we prove that X1 satisfies the required convergence. By the above construction and hypothesis the sequence ˛m ..jXn˛m  E Fn X1 j; Fn /n2N /m2N

is an L1 -bounded submartingale. Applying Lemma V.2.9 in [24] to this sequence yields ˛m ˛m lim sup jXn˛m  E Fn X1 j D sup lim jXn˛m  E Fn X1 jD0

n!1 m2N

m2N n!1

a.s.

Recall that jXn˛  Xnˇ j  C.!/j˛  ˇj and ˇ ˛ j  C.!/j˛  ˇj  X1 jX1

for all ˛; ˇ 2 Œ0; 1. By the Lipschitz property of X1 and Jensen’s inequality, we have the estimation ˛ ˇ ˛ ˇ jE Fn X1  E Fn X1 j  E Fn jX1  X1 j  E Fn C j˛  ˇj ˛ is Lipschitz on for all ˛; ˇ 2 Œ0; 1. Consequently the mapping ˛ 7! Xn˛  E Fn X1 Œ0; 1. So we conclude that ˛m ˛ sup jXn˛m  E Fn X1 j D sup jXn˛  E Fn X1 j

m2N

˛2Œ0;1

On the Integration of Fuzzy Level Sets

31

Finally we get ˛ ˛m lim sup jXn˛  E Fn X1 j D lim sup jXn˛m  E Fn X1 j

n!1 ˛2Œ0;1

n!1 m2N

˛m D sup lim jXn˛m  E Fn X1 j D 0 a.s. m2N n!1

 The above results lead to the study of integrand martingale-submartingalepramarts in a more general context.

References 1. Artstein Z, Hansen JC (1985) Convexification in limit laws of random sets in Banach spaces. Ann Probab 13(1):307–309 2. Castaing C (1970) Quelques résultats de compacité liés à l’intégration. C R Acad Sci Paris Sér A-B 270:A1732–A1735. Actes du Colloque d’Analyse Fonctionnelle de Bordeaux (Univ. Bordeaux, 1971), pp 73–81. Bull. Soc. Math. France, Mém. No. 31–32. Soc. Math. France, Paris (1972) 3. Castaing C (2011) Some various convergence results for normal integrands. In: Kusuoka S, Maruyama T (eds) Advances in mathematical economics, vol 15. Springer, Tokyo, pp 1–26 4. Castaing C, Ezzaki F, Lavie M, Saadoune M (2011) Weak star convergence of martingales in a dual space. In: Hudzik H, Lewicki G, Musielak J, Nowak M, Skrzypczak L (eds) Function spaces IX. Banach center publications, vol 92. Polish Academy of Sciences, Institute of Mathematics, Warsaw, pp 45–73 5. Castaing C, Hess C, Saadoune M (2008) Tightness conditions and integrability of the sequential weak upper limit of a sequence of multifunctions. In: Kusuoka S, Yamazaki A (eds) Advances in mathematical economics, vol 11. Springer, Tokyo, pp 11–44 6. Castaing C, Raynaud de Fitte P (2013) Law of large numbers and ergodic theorem for convex weak star compact valued Gelfand-integrable mappings. In: Kusuoka S, Maruyama T (eds) Advances in mathematical economics, vol 17. Springer, Tokyo, pp 1–37 7. Castaing C, Raynaud de Fitte P, Valadier M (2004) Young measures on topological spaces. With applications in control theory and probability theory. Mathematics and its applications, vol 571. Kluwer Academic, Dordrecht 8. Castaing C, Valadier M (1977) Convex analysis and measurable multifunctions. Lecture notes in mathematics, vol 580. Springer, Berlin/New York 9. de Blasi FS, Tomassini L (2011) On the strong law of large numbers in spaces of compact sets. J Convex Anal 18(1):285–300 10. Feron R (1976) Ensembles aléatoires flous. C R Acad Sci Paris Sér A-B 282(16):Aiii, A903–A906 11. Fitzpatrick S, Lewis AS (2006) Weak-star convergence of convex sets. J Convex Anal 13(3–4):711–719 12. Hiai F (1984) Strong laws of large numbers for multivalued random variables. In: Multifunctions and integrands (Catania, 1983). Lecture notes in mathematics, vol 1091. Springer, Berlin, pp 160–172 13. Hiai F, Umegaki H (1977) Integrals, conditional expectations, and martingales of multivalued functions. J Multivar Anal 7(1):149–182

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Adv. Math. Econ. 19, 33–55 (2015)

A Theory for Estimating Consumer’s Preference from Demand Yuhki Hosoya

Abstract This study shows that if the estimate error of a demand function satisfying the weak axiom of revealed preference is sufficiently small with respect to local C 1 topology, then the estimate error of the corresponding preference relation (which is possibly nontransitive, but uniquely determined from demand function, and transitive under the strong axiom) is also sufficiently small. Furthermore, we show a similar relation for the estimate error of the inverse demand function with respect to the local uniform topology. These results hold when the consumption space is the positive orthant, but are not valid in the nonnegative orthant. Keywords Demand function • Inverse demand function • Integrability theory • Closed convergence topology • Uniform convergence topology • C 1 convergence topology

Article type: Research Article Received: January 9, 2014 Revised: November 14, 2014

JEL Classification: D11. Mathematics Subject Classification (2010): 91B16, 62P20. Y. Hosoya () Department of Economics, Kanto-Gakuin University, Kanazawa-ku, Yokohama-shi, Kanagawa-ken, Japan e-mail: [email protected] 33

34

Y. Hosoya

1 Introduction In economics, estimating a preference relation from observed data is very important for conducting welfare analysis. Both revealed preference theory and integrability theory approach this problem, and share the same key idea. Because purchase behaviors are observed, the demand function is much easier to estimate than the preference relation. Hence, if there exists a method to calculate the preference relation from the demand function, then the difficulty of estimating the preference relation decreases. In the nineteenth century, Antonelli [2] presented a sufficient condition for the local existence of a utility function. Pareto [10] also considered this problem, before Samuelson [14] connected their classical conditions to the symmetry of the Slutsky matrix of the demand function. The general existence of the corresponding preference of the demand function obeying the strong axiom of revealed preference was proved by Richter [13] and Afriat [1]. For computations, Hurwicz and Uzawa [7] presented a method of constructing a utility function from a demand function. Kim and Richter [9] and Quah [12] extended the result of Richter [13] for demand functions obeying the weak axiom of revealed preference. This study is based on the work of Hosoya [6], who presented a method of constructing a preference relation for a smooth demand function with the weak axiom, whose demand function is exactly the same as the original demand function. This preference relation is complete, but may be nontransitive.1 However, the transitivity is revived when this demand function obeys the strong axiom. Moreover, this study shows that such a preference is “unique”, i.e., there is no other usual (that is, complete, continuous, and “p-transitive”, which is defined later) preference relation corresponding to the same demand function. Let P W f 7!%f denote this mapping. Thus, if we obtain an estimate f of the demand function, then we simultaneously obtain an estimate %f D P .f / of the corresponding preference relation. However, the actual estimation process is always associated with the estimate error. If the estimator uses a sophisticated econometric method, then the estimate error of the demand function may become small. However, it remains unknown whether the estimate error of the corresponding preference relation is also reduced. To deal with this problem, we should first clarify the meaning of the term “small”. In general, economists use the closed convergence topology for the space of the preference relations.2 Hence, the sentence “the estimate error of the preference relation is small” means that the estimated preference is within a smallneighborhood

1

Therefore, in this paper the notion of a “preference relation” does not include the transitivity requirement.

2 This topology was induced by Kannai [8]. For a detailed treatment of this topology, see Hildenbrand [5].

A Theory for Estimating Consumer’s Preference from Demand

35

of the true preference with respect to the closed convergence topology. So, what is the topology induced in the space of the demand functions? If the topology is inappropriate, then the estimate error of the corresponding preference relation does not become small. Hence, our problem is the following: what topology on the space of the demand functions makes the function P continuous? This study answers the above question. If .fk /k is a sequence of demand functions that converges to f with respect to the local C 1 topology, then .%fk /k also converges to %f with respect to the closed convergence topology. In addition, we achieve the following result: if .gk /k is a sequence of inverse demand functions that converges to g uniformly on any compact set, then the corresponding sequence of preference relations converges with respect to the closed convergence topology. The reason the topology of the demand function must be finer than that of the inverse demand function is very complicated. We discuss this further in the remarks on our theorems. We obtain results for the consumption space RnCC . However, many applied studies actually assume that the consumption space is RnC . Hence, it is preferable that our results still hold when the consumption space is RnC , but this is not the case. In fact, there exists a usual demand function that does not correspond to any transitive and continuous preference on RnC , though it is smooth and satisfies the strong axiom. We construct such an example. In Sect. 2, we explain some basic results that are needed for our arguments. The first subsection summarizes the results of Hosoya [6]. The second subsection discusses an interpretation of the topologies used in this study. In the last subsection, we explain the basic results of the ordinary differential equation and derive an inequality. The main results are presented in Sect. 3. We present and interpret two theorems. The proofs of all results are given in the appendix. In Sect. 4, we present an example that shows our results cannot be extended to the case where the consumption space is RnC . Section 5 gives our conclusions.

2 Preliminaries Unless otherwise stated, we assume that the consumption space  is RnCC D fx 2 Rn jx i > 0; i D 1; : : : ; ng, where n  2.3

The superscript notation x i means the i -th coordinate of x, and f i means the i -th coordinate of f.

3

36

Y. Hosoya

2.1 Basic Result Let A  RnCC  RCC . We call a function f W A !  a demand function if it satisfies homogeneity of degree zero and Walras’ law.4 Note that we do not assume that the domain A of f is equal to RnCC  RCC , because the consumption space is open, and thus the budget set is not compact. If A is open and f is C 1 -class, define the following matrix: Sf .p; m/ D Dp f .p; m/ C Dm f .p; m/f T .p; m/; where f T .p; m/ denotes the transpose of f .p; m/. This matrix-valued function is called the Slutsky matrix of f . Then, f satisfies the rank condition if the rank of Sf .p; m/ is always n  1. We say f obeys the weak axiom if q
f .p; m/  w; f .p; m/ ¤ f .q; w/ ) p
f .q; w/ > w: F denotes the set of all demand functions f that are onto and C 1 -class, and satisfies both the rank condition and the weak axiom. The following proposition was proved by Hosoya [6]. Proposition 1. Choose any f 2 F . Then, there uniquely exists a function g W  ! RnCC such that g n .x/  1 and, for all x 2 , x D f .g.x/; g.x/
x/: Moreover, g is C 1 -class. A function g W  ! RnCC is called an inverse demand function of f if x D f .g.x/; g.x/
x/ for any x 2 . Then, the above proposition states that if f 2 F , there uniquely exists an inverse demand function g with gn  1, and g is C 1 -class. Let P1 denote the mapping f 7! g. Then, for any demand function f 2 F , P1 .f / is the unique function g W  ! RnCC such that g n .x/  1 and x D f .g.x/; g.x/
x/. Next, choose any relation % 2 .5 We use the notation x % y to represent .x; y/ 2%. Likewise, x  y denotes .x; y/ 2% and .y; x/ …%, and x  y denotes .x; y/ 2% and .y; x/ 2%. Now, define 4

That is, f .ap; am/ D f .p; m/; 8a > 0; p
f .p; m/ D m;

for any .p; m/ 2 RnCC  RCC . 5

Although we do not restrict this binary relation to any class, any relation appearing later is complete and “p-transitive”, which are defined in Proposition 2.

A Theory for Estimating Consumer’s Preference from Demand

37

f % .p; m/ D fx 2 jp
x  m and 8y 2 ; p
y  m ) x % yg: The relation f % is called the demand relation of %. If f % is single-valued, then it is called the demand function of %.6 Now, choose a function g W  ! RnCC and consider the following ordinary differential equation: yP D .g.y/
x/v  .g.y/
v/x; y.0/ D x: Let y.tI x; v/ denote the nonextendable solution of the above equation. Let w.x; v/ D .v
x/v.v
v/x, and t.x; v/ D minft  0jy.tI x; v/
w.x; v/ D 0g. Then, we can prove that t.x; v/ is well-defined and if x is not proportional to v, it indicates a unique t such that y.tI x; v/ is proportional to v. Hence, y.t.x; v/I x; v/ D cv for some constant c > 0. If x is proportional to v, then y.tI x; v/  x, and thus . Let ug .x; v/ denote t.x; v/ D 0 and y.t.x; v/I x; v/ D cv for the constant c D kxk kvk such a constant c. Define the relation %g  2 as x %g v , ug .x; v/  1: The following proposition was proved by Hosoya [6]. Proposition 2. If g W  ! RnCC is C 1 -class, then t.x; v/, and thus ug .x; v/; %g , is well-defined. Moreover, ug and %g satisfy the following properties: – Completeness. For any .x; v/ 2 2 , x %g v or v %g x. – P-transitivity. For any x; y; z 2 2 with dim.spanfx; y; zg/  2, x %g y and y %g z imply x %g z. – Continuity. ug is continuous on 2 and %g is closed in 2 . – Strong monotonicity. x g v if x ¤ v and x i  v i for any i D 1; : : : ; n. – Representation. For any x; v 2  and y; z 2 spanfx; vg \ , y %g z , ug .y; v/  ug .z; v/: Moreover, if g D P1 .f / for some f 2 F , then7

Clearly, f % satisfies the homogeneity of degree zero. In addition, f % satisfies Walras’ law if % is locally nonsatiated, i.e., for any x 2  and any neighborhood U of x, there exists y 2 U such that y  x.

6

g

7 This implies x D f % .g.x/; g.x/
x/ for any x 2 . This result can be shown even if g is g not included in the range of P1 . In fact, x 2 f % .g.x/; g.x/
x/ for any x 2  if and only if T n w Dg.x/w  0 for any x 2  and w 2 R with g.x/
w D 0. However, if g D P1 .f / for some f 2 F , this condition holds automatically. See Theorem 1 and Proposition 1 of Hosoya [6] for more detailed arguments.

38

Y. Hosoya g

f % D f; and, for any % 2 with f % D f , if it satisfies completeness, p-transitivity, and continuity, then %D%g . Let G denote the set of all C 1 -class function g W  ! RnCC and P2 denote the mapping g 7!%g . We call %g D P2 .P1 .f // the “corresponding” preference relation of f . The reason for this name is as follows. First, in economics, the “preference relation” usually means the complete and transitive binary relation. If % is transitive, then it is clearly p-transitive. Therefore, if % is complete, transitive, and continuous binary relation and f % D f , then %D%g by the last result of Proposition 2. Second, Hosoya [6] showed the following additional result. Suppose that %g D P2 .P1 .f //. Then, %g is transitive if and only if f satisfies the strong axiom. If f D f % for some complete and transitive binary relation %, then it satisfies the strong axiom. This fact show that if f D f % for some complete and transitive (and not necessarily continuous) binary relation %, then %g is the unique complete, transitive, and continuous binary relation that satisfies f D f % . In other words, if %g is nontransitive, then there is no transitive binary relation % that derives f . Third, this relation %g must be transitive whenever  D R2CC . Last, this relation %g is easy to interpret. Since g D P1 .f / is an inverse demand function, g.x/ is a price vector when x is bought. The reader can be easily verified that yP in the above differential equation is orthogonal to g.y/. Therefore, the curve y is tangent to the budget line in the plane spanfx; vg at each point, and can thus be interpreted as the indifference curve passing through x. Then, our definition says that “x is preferred to v if and only if the indifference curve y passes over v”. This interpretation is very natural. Now, we can prove that ug has the following properties: ug .x; v/ > 1 , ug .v; x/ < 1; ug .x; av/ D

1 g u .x; v/; a

ug .x; v/  1 if x  v: The first property comes from the representation property of ug . In fact, because x; v 2 spanfx; vg and ug .v; v/ D 1, ug .x; v/ > 1 , ug .x; v/ > ug .v; v/ , v 6%g x , ug .v; x/ < 1: The second property is a direct consequence of the definition of ug . The final property arises from strong monotonicity.

A Theory for Estimating Consumer’s Preference from Demand

39

2.2 Some Properties of Topologies Let P denote the set of all relations % 2 that are continuous, i.e., closed in 2 . On P, we introduce a topology, known as the closed convergence topology. For any sequence .%k /k of P, define the following two sets Limsupk %k and Liminfk %k . .x; v/ 2 Limsupk %k if and only if for any neighborhood U of .x; v/ and any k0 2 N, there exists k  k0 such that %k \U ¤ ;. .x; v/ 2 Liminfk %k if and only if for any neighborhood U of .x; v/, there exists k0 2 N such that %k \U ¤ ; for any k  k0 . By construction, we have Liminfk %k  Limsupk %k . If Liminfk %k D Limsupk %k D% for some %2 P, then we write Limk %k D%, and the topology induced by this definition of the limit is called the closed convergence topology. Note that because Liminfk %k  Limsupk %k , we can prove Limk %k D% by showing that Limsupk %k % Liminfk %k . Next, consider the space C.X / of all continuous functions on the locally compact Hausdorff topological space X into Rm . We say the sequence .fk /k of C.X / converges to f uniformly on any compact set if for any compact set K  X , lim sup kfk .x/  f .x/k ! 0:

k!1 x2K

Then, we can show the following fact. .fk /k converges to f uniformly on any compact set if and only if .fk /k converges to f locally uniformly, that is, for any x 2 X , there exists a neighborhood U of x such that lim sup kfk .x/  f .x/k ! 0:

k!1 x2U

As X is locally compact, if .fk /k converges to f uniformly on any compact set, then it converges locally uniformly. Conversely, suppose .fk /k converges to f locally uniformly. Choose any compact set K and any x 2 K, and let Ux denote the neighborhood of x that appears in the definition of local uniform convergence. Then, .Ux /x2K covers K, and thus there exists x1 ; : : : ; xM 2 K such that K  [M i D1 Uxi . Then, sup kfk .x/  f .x/k  max sup kfk .x/  f .x/k ! 0

x2K

i

x2Uxi

as k ! 1.

2.3 On Differential Equations In this subsection, we note several properties of the solution of ordinary differential equations.

40

Y. Hosoya

First, consider the following ordinary differential equation: xP D f .t; x; y; z/; x.t0 / D y; where f is defined on some open set A  RRm Rm Rk that includes .t0 ; y; y; z/, and is C 1 -class. Then, there uniquely exists a function x.t/ that is defined on an .t/ D f .t; x.t/; y; z/, and open interval a; bŒ with a < t0 < b, x.t0 / D y, dx dt is nonextendable: that is, there is no other function y.t/ that is defined on c; d Œ with a; bŒ¨c; d Œ, y.t/ D x.t/ if t 2a; bŒ, and dy .t/ D f .t; y.t/; y; z/.8 We call dt this function x the nonextendable solution under .t0 ; y; z/. The following result is well-known: suppose ft0 g  C  f.y; z/g  A and C is compact. Let the domain of the nonextendable solution x under .t0 ; y; z/ be a; bŒ. If b < C1, then there exists t  < b such that if t   t < b, then x.t/ … C .9 Now, let x.tI y; z/ have the following property: for any .y; z/ with .t0 ; y; y; z/ 2 A, x.y;z/ W t 7! x.tI y; z/ is the nonextendable solution of the above differential equation under .t0 ; y; z/. We also call the function x itself the nonextendable solution. It is well-known that the domain of x is an open set in R  Rm  Rk and that x is continuous in .t; y; z/.10 R t The following result was given in the 2nd editionb ofatPontryagin [11]: if u.t/   1/ for any t 2 Œ0; c. 0 .au.s/ C b/ds for any t 2 Œ0; c, then u.t/  a .e However, the 2nd edition of this book has not been translated to English. Therefore, we introduce a short sketch of the proof of this inequality. Let u0 .t/ D u.t/ and ui .t/ D

Z

t 0

.aui 1 .s/ C b/ds:

That is, ui is the Picard’s iteration of the following integral equation: v.t/ D

Z

t 0

.av.s/ C b/ds:

Note that the solution of this equation is v.t/ D ab .e at  1/. By mathematical induction, we can show that ui .t/  ui 1 .t/ for any i , and ui ! v uniformly. Hence, we have ui .t/  v.t/ for any i , and especially, u.t/  v.t/.

8

Note that a might be 1 and b might be C1.

9

See Ch.8 of Smale and Hirsch [15] for detailed arguments.

10

See Ch.5 of Hartman [4] for detailed arguments.

A Theory for Estimating Consumer’s Preference from Demand

41

3 Main Results We claim the following result. Theorem 1. If .gk /k converges to g uniformly on any compact set, and all gk and g are C 1 -class, then %gk converges to %g with respect to the closed convergence topology. Theorem 2. Suppose that .fk /k is a sequence of F and converges to f 2 F in the sense of local C 1 : that is, for any .p; m/ 2 dom.f /, there exists a neighborhood U of .p; m/ such that U  dom.fk / for any sufficiently large k, and lim

sup kfk .q; w/  f .q; w/k D 0;

k!1 .q;w/2U

and lim

sup kDfk .q; w/  Df .q; w/k D 0:

k!1 .q;w/2U

Then, .gk /k D .P1 .fk //k converges to g D P1 .f / uniformly on any compact set. Combining these theorems, we obtain the following result. Theorem 3. Suppose .fk /k and f satisfy the same property as in Theorem 2. Then the relation .P2 .P1 .fk ///k converges to P2 .P1 .f // with respect to the closed convergence topology. Remarks. Theorem 1 asserts that if the estimate error on the inverse demand function is small with respect to the local uniform topology, then the estimate error of the preference is also small. It is known that local uniform topology is metrizable by the following metric: .g1 ; g2 / D

1 X 1 arctan. sup kg1 .y/  g2 .y/k/: m 2 y2Œ 1 ;mn mD1 m

Similarly, Theorem 3 asserts that if the estimate error on the demand function is small with respect to local C 1 topology, then the estimate error of the preference is also small. Why does Theorem 2 require C 1 convergence? Answering this question is rather difficult, but we attempt to interpret this requirement. Suppose .hk /k is a sequence of continuous functions that converges uniformly to h. Moreover, suppose all h; hk are bijective. Our question is the following: is .h1 k /k equicontinuous? If it is equicontinuous, then by using Ascoli-Arzela theorem, we can show that 1 .h1 uniformly. However, we could not ensure that .h1 k /k converges to h k /k is equicontinuous unless .hk /k converges to h with respect to C 1 . Readers will see

42

Y. Hosoya

that, in the proof of Theorem 2, Lemma 4 ensures this equicontinuity, though its proof is rather complicated. The proof of this fact is closely related to the proof of the inverse function theorem. In fact, Lemma 4 can be seen as an extension of inverse function theorem. To prove Theorem 1, we separate the proof into two parts. In the first, we prove that Limk %gk D%g if .ugk /k converges to ug uniformly on any compact set. This result can be verified easily. We should mention the following fact: if g satisfies the integrability condition (or equivalently, if the Slutsky matrix Sf .p; m/ is always g g symmetric), then ug .x; z/ D uug.x;v/ for any x; z; v 2  and uv W x 7! ug .x; v/ .z;v/ is a C 1 -class utility function representing %g .11 Therefore, this result implies that g g Limk %gk D%g if the sequence of utility functions .uv k /k converges to uv uniformly on any compact set. In the second part, we prove that if .gk /k converges to g uniformly on any compact set, then .ugk /k converges to ug uniformly on any compact set. To prove this result, we will use several deep results for ordinary differential equations. The main problem to be treated is the following: let V  2 be compact, and define y.tI x; v/; yk .tI x; v/; t.x; v/ and tk .x; v/ as in Sect. 2.1. Then, is t.x; v/ included in the domain of yk .
I x; v/ for any .x; v/ 2 V and sufficiently large k? Or, is tk .x; v/ included in the domain of y.
I x; v/? Note that t is not necessarily continuous at .x; v/ when x is proportional to v. To answer this problem, we should use the inequality proved in Sect. 2.3.

4 The Case of Corner Solutions We have assumed that the consumption space  is RnCC . If  is RnC , can our results in both Hosoya [6] and this paper be extended to this case? Let F  be the all function f W RnCC  RCC ! RnC such that it satisfies continuity, homogeneity of degree zero, Walras’ law, and the weak axiom, RnCC  f .RnCC  RCC /, and if f .p; m/ 2 RnCC , then f is C 1 -class around .p; m/ and Sf .p; m/ has rank n  1. Let R.f / be the restriction of f into f 1 .RnCC /. Then, R.f / 2 F , and thus there uniquely exist a complete, p-transitive, and continuous g preference relation %g D P2 .P1 .R.f /// such that f % D R.f /. Clearly, if .fk /k is a sequence in F  that converges to f 2 F  with respect to the local C 1 topology,12 then we can easily show that .R.fk //k also converges to R.f / with respect to the local C 1 topology. Thus, we have .P2 .P1 .R.fk ////k converges to P2 .P1 .R.f ///. Then, the next question naturally arises: does there uniquely exist a

11

See Theorem 1 of Hosoya [6].

That is, .fk /k converges to f uniformly on any compact set, and for any .p; m/ with f .p; m/ 2 RnCC , there exists a neighborhood U of .p; m/ such that .Dfk /k converges to Df uniformly on U. 12

A Theory for Estimating Consumer’s Preference from Demand

43 f

complete, p-transitive, and continuous preference %f  .RnC /2 such that f % D f ? Moreover, is the operator f 7!%f continuous with respect to the local C 1 topology? However, the answer to this question is negative. In fact, we can construct a preference relation % .R2CC /2 that is represented by a C 1 -class utility function, and two different indifference curves of % have the same limit point. In this case, any extension of % into the complete binary relation of R2C is either not continuous or not transitive. Note that this preference has the C 1 -class demand function f % 2 F \ F. Example 1. At first, we consider the following function: h1 .c/ D

(

1

e  c2

(if c > 0,)

0

(otherwise.)

It is well-known that this function is C 1 -class, increasing on Œ0; C1Œ, and h1 .c/ ! 1 if c ! 1. Next, let h1 .1  c/ ; h1 .1/  h3 .c/ D tan. h1 .c  2//; 2 h2 .c/ D 1 

h.c/ D h2 .c/ C h3 .c/: Then, h is a C 1 -class nondecreasing function, h.c/ D 1 if c 2 Œ1; 2, increasing on Œ0; C1ŒnŒ1; 2, h.c/ # 0 if c # 0, and h.c/ " 1 if c " 1. Next, let x; y > 0 and consider the following equation: 1

1

1

1

1

.x 1C c C y 1C c /1C c D h.c/: Because the derivative of the left-hand side of this equation with respect to c is negative, we have the solution c.x; y/ of this equation is unique. By the implicit function theorem, we have c.x; y/ is C 1 -class. As the left-hand side is increasing in both x and y, we have c.x; y/ is increasing in both x and y. Let .x; y/ % .z; w/ , c.x; y/  c.z; w/: Then, the indifference curve of % is 1

1

1

1

1

1

L.c/ D f.x; y/ 2 R2CC jx 1C c C y 1C c D h.c/ 1C c g;

44

Y. Hosoya

and thus we can verify that it is strictly convex toward the origin and has non-zero Gaussian curvature.13 For any p; q; m > 0, define .x.t/; y.t// D .t; mpt q /. Then d c.x.t/; y.t// dt

d is positive if t is sufficiently small and dt c.x.t/; y.t// is negative if % m=p  t is sufficiently small. Therefore, we have f is a single-valued C 1 -class mapping from R2CC  RCC onto R2CC , that is, f % 2 F  \ F . However, for any c 2 Œ1; 2, .0; 1/ and .1; 0/ are the limit points of L.c/. This indicates that there is no continuous and transitive extension of % on R2C . In fact, if % is such an extension, we have

1 1 1 1 . ; /  .0; 1/  . 3=2 ; 3=2 /; 4 4 2 2 1 1 but c. 14 ; 14 / < c. 23=2 ; 23=2 /, a contradiction.14

Figure 1 illustrates the above example. Note that, by Proposition 1, if some transitive and continuous preference %0 has the same demand function f % , then %0 D%. This example indicates a possible unconformity between usual demand functions and the transitivity assumption of continuous preferences on RnC . Although the demand function is C 1 -class and satisfies the strong axiom, there may be no complete, transitive, and continuous binary relation that derives this demand function. Fig. 1 Illustration of example

y

1

L(2)

L(1) x

13 14

This condition assures the differentiability of the demand function. See Debreu [3].

This % has a unique upper semi-continuous and transitive extension. However, we cannot state whether this property is general or not.

A Theory for Estimating Consumer’s Preference from Demand

45

5 Conclusion We have shown that if the estimate error of a demand function is sufficiently small with respect to the local C 1 topology, then the estimate error of the corresponding preference relation is also small with respect to the closed convergence topology. Moreover, we showed that if the estimate error of an inverse demand function is sufficiently small with respect to the local uniform topology, then the estimate error of the corresponding preference relation is also small. Furthermore, we gave an example that indicates our results cannot be extended in the case of a closed consumption space. Three future tasks remain. At first, we defined %g from an ordinary differential equation. But if f violates the rank condition, then g cannot become a continuous single-valued function.15 However, we might be able to define %g by considering differential inclusion. Secondly, we did not treat the case in which f is a multig valued function. In some cases, f % might be multi-valued, and thus this should be considered. Thirdly, even when the consumption space is the nonnegative orthant, there might be an appropriate class of preferences such that f 7!%f is a one-to-one mapping for any F  .

Appendix 1: Proof of Theorem 1 We at first introduce a lemma. Lemma 1. Let .%k /k be a sequence on P and %2 P. Suppose x %k v , uk .x; v/  1 x % v , u.x; v/  1 for some real-valued function uk ; u on 2 . Assume u; uk are continuous and16 uk .x; v/ > 1 , uk .v; x/ < 1; u.x; v/ > 1 , u.v; x/ < 1; uk .x; av/ D

1 1 uk .x; v/; u.x; av/ D u.x; v/; a a

uk .x; v/  1 and u.x; v/  1 if x  v: Then, Limk %k D% with respect to the closed convergence topology if .uk /k converges to u uniformly on any compact set.

15

See Samuelson [14] or Hurwicz and Uzawa [7].

16

Note that under these conditions, %; %k must satisfy completeness and continuity.

46

Y. Hosoya

Proof. Suppose .uk /k converges to u uniformly on any compact set, and choose any .x; v/ 2 Limsupk %k . Then, for any neighborhood U of .x; v/ and any k0 2 N, there exists k  k0 such that %k \U ¤ ;. Therefore, there exists an increasing sequence .k.m// such that uk.m/.xm ; vm /  1 and k.xm ; vm /  .x; v/k  m1 . By the assumption of uniform convergence, we have juk.m/.xm ; vm /  u.xm ; vm /j ! 0. Because u is continuous, u.xm ; vm / ! u.x; v/, and thus uk.m/.xm ; vm / ! u.x; v/. As uk.m/.xm ; vm /  1, we have u.x; v/  1, and thus Limsupk %k %. Next, under the same assumption, choose any .x; v/ 2%. Then, we have u.x; v/  1. This implies u.x; .1  m1 /v/ > 1 for any m. Now, take any neighborhood U of .x; v/. Then, there exists m > 0 such that .x; .1  m1 /v/ 2 U . Because uk .x; .1  m1 /v/ ! u.x; .1  m1 /v/ > 1, there exists k0 2 N such that .x; .1  m1 /v/ 2%k for any k  k0 . Hence, we have % Liminfk %k . Therefore, we have Limk %k D%.  Hence, it suffices to show that .ugk /k converges to ug uniformly on any compact set. The next lemma was proved in Hosoya [6]. Thus, we omit the proof of this result.

Lemma 2. Let g W  ! RnCC be C 1 -class, and y.tI x; v/ be the nonextendable solution of the following problem: yP D .g.y/
x/v  .g.y/
v/x; y.0/ D x; where x; v 2 . Define w.x; v/ D .v
x/v  .v
v/x. Then, for any z 2 spanfx; vg, z
w.x; v/ D 0 if and only if z is proportional to v. Let a; bŒ be the domain of y.
I x; v/. Then, t.x; v/ D minft  0jy.tI x; v/
w.x; v/  0g is well-defined. Moreover, there exists a continuous function y1 .x; v/; y2 .x; v/ such that y1 .x; v/ D y2 .x; v/ D x if x is proportional to v, and y.t.x; v/I x; v/ 2 Œy1 .x; v/; y2 .x; v/ for any .x; v/ 2 2 . Let A.x; v/ D kxkkv 

v
x xk: kxk2

Then, w.x; v/ is orthogonal to v, w.x; v/ ¤ 0 if and only if x is not proportional to v, and if w.x; v/ ¤ 0, yP
w.x; v/ D A.x; v/2 .g.y/
v/ > 0: Hence, y.tI x; v/
w.x; v/ is increasing in t, and thus t.x; v/ is a unique t such that y.tI x; v/
w.x; v/ D 0. Lemma 3. Suppose .gk /k converges to g uniformly on any compact set. Then, .ugk /k converges to ug uniformly on any compact set.

A Theory for Estimating Consumer’s Preference from Demand

47

Proof. Let y.tI x; v/ be the nonextendable solution of the following equation: yP D .g.y/
x/v  .g.y/
v/x; y.0/ D x; and yk .tI x; v/ be the nonextendable solution of the following equation: yP D .gk .y/
x/v  .gk .y/
v/x; y.0/ D x: Let y.
I x; v/ be defined on a.x; v/; b.x; v/Œ. We separate this proof into four steps. Step 1: fix any .x; v/ 2 2 and c 20; b.x; v/Œ. Then, there exists a compact neighborhood V of .x; v/ such that y.
I y; z/ is defined on Œ0; c for any .y; z/ 2 V . The set fy.tI y; z/jt 2 Œ0; c; .y; z/ 2 V g is a compact set of , and thus there exists a compact set C   such that the above set is included in the interior of C . We shall prove that, for sufficiently large k, yk .tI y; z/ is defined for any .t; y; z/ 2 Œ0; c  V , yk .tI y; z/ 2 C , and kyk .tI y; z/  y.tI y; z/k 

kgk  gk 2M kykkzkt .e  1/; M

(1)

where M > 0 is a Lipschitz constant of g on C and kgk gk D supw2C kgk .w/ g.w/k. First, yk .tI y; z/  y.tI y; z/ D .yk .tI y; z/  y/  .y.tI y; z/  y/ Z t ŒyPk .sI y; z/  y.sI P y; z/ds: D 0

Hence, if t 2 Œ0; c is in the domain of yk .
I y; z/ and yk .sI y; z/ 2 C for any s 2 Œ0; t, then we have kyk .tI y; z/  y.tI y; z/k Z t kgk .yk .sI y; z//  g.y.sI y; z//k
2kykkzkds   C

Z

Z

0

t

kg.yk .sI y; z//  g.y.sI y; z//k
2kykkzkds

0

0

t

kgk .yk .sI y; z//  g.yk .sI y; z//k
2kykkzkds

 2kykkzk  2kykkzk

Z

Z

t

Œkg.yk .sI y; z//  g.y.sI y; z//k C kgk  gkds

0 t 0

.M kyk .sI y; z/  y.sI y; z/k C kgk  gk/ds;

(2)

48

Y. Hosoya

which implies equation (1).17 Hence, if k is sufficiently large (and thus, kgk  gk is sufficiently small), then kyk .tI y; z/  y.tI y; z/k  ı, where ı > 0 is sufficiently small so that the following inequality holds: inf

w…C;t 2Œ0;c;.y;z/2V

kw  y.tI y; z/k  2ı:

Now, define t  D supfs 2 Œ0; cj8 2 Œ0; s; yk .I y; z/ 2 C g. Then yk .tI y; z/ 2 C for any t 2 Œ0; t  Œ. If t  is not included in the domain of yk .
I y; z/, then there exists tO < t  such that yk .tI y; z/ … C for any t 2 ŒtO; t  Œ, a contradiction. Hence, t  is in the domain of yk .
I y; z/ and yk .t  I y; z/ 2 C . It suffices to show that t  D c. Suppose t  < c. Then, the above arguments implies kyk .t  I y; z/  y.t  I y; z/k  ı, and thus yk .t  I y; z/ is included in the interior of C . Hence, yk .
I y; z/ is defined on Œt  ; t  C" and yk .Œ0; t  C"I y; z/  C for sufficiently small " > 0, a contradiction. This completes the proof of our claim. Step 2: define w.x; v/ as in Lemma 2, and t.x; v/ (resp. tk .x; v/) as the minimum constant t  0 such that y.tI x; v/ (resp. yk .tI x; v/) is proportional to v. By Lemma 2, if x is not proportional to v, then t D t.x; v/ if and only if y.tI x; v/
w.x; v/ D 0 and t D tk .x; v/ if and only if yk .tI x; v/
w.x; v/ D 0. Also, again by Lemma 2, t > t.x; v/ if and only if y.tI x; v/
w.x; v/ > 0 and t > tk .x; v/ if and only if yk .tI x; v/
w.x; v/ > 0. Note that w.x; v/ D 0 and t.x; v/ D tk .x; v/ D 0 if and only if x is proportional to v. Step 3: choose any compact subset V of 2 . Fix any " > 0. Let W D f.x; v/ 2 V jky1 .x; v/  y2 .x; v/k < kvk"g, where y1 ; y2 are as in Lemma 2. Because y1 ; y2 is continuous, W is open in V , and thus V 0 D V n W is compact. By Lemma 2, if .x; v/ 2 W , then both yk .tk .x; v/I x; v/ and y.t.x; v/I x; v/ are included in Œy1 .x; v/; y2 .x; v/, and thus, kyk .tk .x; v/I x; v/  y.t.x; v/I x; v/k  kvk": Note that 1 1 y.t.x; v/I x; v/
v; ugk .x; v/ D yk .tk .x; v/I x; v/
v: kvk2 kvk2 (3) Hence, if .x; v/ 2 W , ug .x; v/ D

jug .x; v/  ugk .x; v/j D

1 j.yk .tk .x; v/I x; v/  y.t.x; v/I x; v//
vj kvk2

kyk .tk .x; v/I x; v/  y.t.x; v/I x; v/k kvk  ": 

17

See Sect. 2.3.

A Theory for Estimating Consumer’s Preference from Demand

49

Meanwhile, suppose .x; v/ 2 V 0 . If x is proportional to v, then y1 .x; v/ D y2 .x; v/ D x, and thus .x; v/ 2 W , a contradiction. Hence, x is not proportional to v. Because t.x; v/ is the unique solution of the equation y.tI x; v/
w.x; v/ D 0, by the implicit function theorem, the function t.x; v/ is continuous on V 0 . Choose any .x; v/ 2 V 0 and any c.x; v/ 2t.x; v/; b.x; v/Œ, and let V.x;v/ be a compact neighborhood of .x; v/ in V 0 such that y.
I y; z/ is defined on Œ0; c.x; v/ and t.y; z/ < c.x; v/ for any .y; z/ 2 V.x;v/ . By step 1, we can take a compact set C.x;v/  RnCC such that (i) y.tI y; z/ 2 C.x;v/ for any .t; y; z/ 2 Œ0; c.x; v/  V.x;v/ , (ii) There exists k0 such that for any k  k0 and .y; z/ 2 V.x;v/ , yk .
I y; z/ is also defined on Œ0; c.x; v/ and yk .tI y; z/ 2 C.x;v/ for any t 2 Œ0; c.x; v/, and (iii) For any k  k0 , Eq. (1) holds for any t 2 Œ0; c.x; v/.

Because V 0 is compact, there is a finite collection f.x1 ; v1 /; : : : ; .xm ; vm /g of V 0 such that [i V.xi ;vi / D V 0 . Hence, it suffices to show that for any i 2 f1; : : : ; mg, there exists ki such that for any k  ki , sup.x;v/2V.x ;v / jugk .x; v/  ug .x; v/j  ". i i For notational convention, we abbreviate V.xi ;vi / as Vi , C.xi ;vi / as Ci , and c.xi ; vi / as ci . Step 4: by definition of Vi , we have t.x; v/ is in the domain of yk if k is sufficiently large. Hence, jugk .x; v/  ug .x; v/j D 

1 j.yk .tk .x; v/I x; v/  y.t.x; v/I x; v//
vj kvk2

1 j.yk .tk .x; v/I x; v/  yk .t.x; v/I x; v//
vj kvk2 C

1 j.yk .t.x; v/I x; v/  y.t.x; v/I x; v//
vj: kvk2

By Eq. (1), 1 jyk .t.x; v/I x; v/  y.t.x; v/I x; v/
vj kvk2  

1 kyk .t.x; v/I x; v/  y.t.x; v/I x; v/k kvk kgk  gk 2M kxkkvkci .e  1/; M kvk

50

Y. Hosoya

where the right-hand side is less than 2" uniformly on Vi if k is sufficiently large. Hence, it suffices to show that, for any sufficiently large k, max

.x;v/2Vi

" 1 j.yk .tk .x; v/I x; v/  yk .t.x; v/I x; v//
vj  : 2 kvk 2

By the Cauchy-Schwarz inequality, it suffices to show that, for any sufficiently large k, max kyk .tk .x; v/I x; v/  yk .t.x; v/I x; v/k 

.x;v/2Vi

kvk" : 2

Recall that y.tI x; v/
w.x; v/ > 0 (resp. yk .tI x; v/
w.x; v/ > 0) if and only if t > t.x; v/. (resp. t > tk .x; v/.) As ci > t.x; v/, we have there exists a positive constant C > 0 such that y.ci I x; v/
w.x; v/  C for any .x; v/ 2 Vi . Hence, for sufficiently large k, yk .ci I x; v/
w.x; v/ > 0 for any .x; v/ 2 Vi . Therefore, ci > tk .x; v/, and thus tk .x; v/ is in the domain of y.
I x; v/. Now, define M1 D

inf

k2N;.x;v/2Vi ;y2Ci

M2 D

min

.x;v/2Vi ;y2Ci

f..gk .y/
x/v  .gk .y/
v/x/
w.x; v/g;

f..g.y/
x/v  .g.y/
v/x/
w.x; v/g:

By Lemma 2, when x is not proportional to v, ..gk .y/
x/v  .gk .y/
v/x/
w.x; v/ D A.x; v/2 .gk .y/
v/ > 0; ..g.y/
x/v  .g.y/
v/x/
w.x; v/ D A.x; v/2 .g.y/
v/ > 0: Because gk ! g uniformly on Ci , we can show that M1 ; M2 > 0. As both y.t.x; v/I x; v/ and yk .tk .x; v/I x; v/ are proportional to v, we have y.t.x; v/I x; v/
w.x; v/ D yk .tk .x; v/I x; v/
w.x; v/ D 0. If t.x; v/  tk .x; v/, then yk .t.x; v/I x; v/
w.x; v/ D 

Z

Z

t .x;v/ tk .x;v/

yPk .tI x; v/
w.x; v/dt

t .x;v/

M1 dt tk .x;v/

D M1 .t.x; v/  tk .x; v//: Hence, we have t.x; v/  tk .x; v/ C

1 yk .t.x; v/I x; v/
w.x; v/: M1

A Theory for Estimating Consumer’s Preference from Demand

51

Similarly, if t.x; v/  tk .x; v/, then y.tk .x; v/I x; v/
w.x; v/ D 

Z

Z

tk .x;v/ t .x;v/

y.tI P x; v/
w.x; v/dt

tk .x;v/

M2 dt t .x;v/

D M2 .tk .x; v/  t.x; v//: Hence, we have t.x; v/  tk .x; v/ 

1 y.tk .x; v/I x; v/
w.x; v/: M2

Now, y.t.x; v/I x; v/ is proportional to v, and thus y.t.x; v/I x; v/
w.y; z/ D 0. Hence, by Eq. (1), for any c > 0, max jyk .t.x; v/I x; v/
w.x; v/j

.x;v/2Vi

D max j.yk .t.x; v/I x; v/  y.t.x; v/I x; v//
w.x; v/j .x;v/2Vi

 max kyk .t.x; v/I x; v/  y.t.x; v/I x; v/kkw.x; v/k .x;v/2Vi

 c; for sufficiently large k. Similarly, max jy.tk .x; v/I x; v/
w.x; v/j

.x;v/2Vi

D max j.yk .tk .x; v/I x; v/  y.tk .x; v/I x; v//
w.x; v/j .x;v/2Vi

 max kyk .tk .x; v/I x; v/  y.tk .x; v/I x; v/kkw.x; v/k .x;v/2Vi

 c; for sufficiently large k. For such k, we have jt.x; v/  tk .x; v/j  M3 D minfM1 ; M2 g. Then, if c > 0 is sufficiently small, kyk .tk .x; v/I x; v/  yk .t.x; v/I x; v/k

Z

tk .x;v/

yPk .tI x; v/dt D

t .x;v/ ˇ ˇZ ˇ ˇ tk .x;v/ ˇ ˇ Œmax k.g.y/
x/v  .g.y/
v/xkdt ˇ ˇ ˇ ˇ t .x;v/ y2Ci

c M3 ,

where

52

Y. Hosoya

 max k.g.y/
x/v  .g.y/
v/xk y2Ci



c M3

kvk" ; 2

which completes the proof.



Lemmas 1 and 3 ensure that the claim of Theorem 1 holds. This completes the proof of Theorem 1. 

Appendix 2: Proof of Theorem 2 Define h.q/ D f .q 1 ; : : : ; q n1 ; 1; q n / and hk .q/ D fk .q 1 ; : : : ; q n1 ; 1; q n /. Then all h; hk W RnCC ! RnCC are invertible18 and, for any q 2 RnCC , there exists a neighborhood U of q such that lim sup khk .r/  h.r/k D 0;

(4)

lim sup kDhk .r/  Dh.r/k D 0:

(5)

k!1 r2U k!1 r2U

Moreover, by the rank condition, we have all Dh.q/; Dhk .q/ are regular.19 Clearly, lim kDhk .q/1  Dh.q/1 k D 0

k!1

for any q 2 RnCC . Next, choose any x 2  and corresponding q 2 RnCC such that h.q/ D x. Then, there exists k0 2 N such that some neighborhood of q is included in the domain of hk for any k  k0 . We now introduce a lemma. Lemma 4. For any sufficiently small ı > 0, there exists M > 0 such that kh.r/  h.q/k  ı or khk .r/  hk .q/k  ı for some k  k0 implies kr  qk  M ı. Moreover, M can be chosen independent to ı. Proof. Let .r/ D Dh.q/.r  q/  .h.r/  h.q//; k .r/ D Dhk .q/.r  q/  .hk .r/  hk .q//:

18

That is, all h; hk are bijective. Injectivity follows from the uniqueness of the inverse demand function. Surjectivity arises from the surjectivity and homogeneity of f and fk . 19

See the mathematical appendix of Samuelson [14], or the proof of Proposition 1 in Hosoya [6].

A Theory for Estimating Consumer’s Preference from Demand

53

Then, by the assumption of convergence of hk , M D 2 supfkDh.q/1 k; kDhk0 .q/1 k; : : :g < C1. Since D.r/ D Dh.q/  Dh.r/ and Dk .r/ D Dhk .q/  Dhk .r/, for any sufficiently small ı > 0, kr  qk  M ı implies that r is included in the domain of hk for any k  k0 and20 supfkDh.q/1 kkD.r/k; kDhk0 .q/1 kkDk0 .r/k; : : :g  Now, let S D fsjksk  M ıg and choose any r 2

1 M

1 : 2

S . Define,

r .s/ D Dh.q/1 .r C .s C q//; rk .s/ D Dhk .q/1 .r C k .s C q//: Then, if s 2 S , kr .s/k  kDh.q/1 kkrk C kDh.q/1 kk.s C q/k M ı C kDh.q/1 kkD.ts C q/kksk 2 1 M ı C M ı  M ı;  2 2



where t 2 Œ0; 1.21 Hence, r .S /  S . Moreover, kr .s1 /  r .s2 /k  kDh.q/1 kk.s1 C q/  .s2 C q/k  kDh.q/1 kkD.s3 C q/kks1  s2 k 

1 ks1  s2 k; 2

where s3 2 Œs1 ; s2 .22 Hence, r is a contraction. By contraction mapping fixed point theorem, there uniquely exists s 2 S such that r .s/ D s, that is, s D Dh.q/1 r C s  Dh.q/1 .h.s C q/  h.q//; and thus h.s C q/ D h.q/ C r. Because h is an injection, we have kh.s C q/  h.q/k  ı implies ksk  M ı. Similarly, we have khk .s C q/  hk .q/k  ı implies  ksk  M ı. This completes the proof of Lemma 4.

Note that y D h.r/ if and only if r D .g1 .y/; : : : ; g n1 .y/; g.y/
y/ and y D hk .r/ if and only if r D .gk1 .y/; : : : ; gkn1 .y/; gk .y/
y/. For any " > 0 there exists

20

See Eq. (5).

21

Use the mean value theorem for k.t / D .t s C q/.

22

Again, use the mean value theorem for k.t / D ..1  t /s1 C t s2 C q/.

54

Y. Hosoya

ı > 0 such that kh.r/  h.q/k  ı or khk .r/  hk .q/k  ı for some k  k0 implies kr  qk  M ı and M ı  ". Let U D fy 2 jkx  yk  ıg. Then for any y 2 U , 1 kgk .y/  gk .x/k  kh1 k .y/  hk .x/k  ":

Hence we conclude that .gk /k is equicontinuous at x. As x is an arbitrary point of , we conclude that .gk /k is equicontinuous. Next, we will show that gk .x/ ! g.x/ pointwise. Let h.q/ D hk .qk / D x. It suffices to show that qk ! q. By assumption, there exists a neighborhood V of q and k0 2 N such that V is included in the domain of hk for any k  k0 . Without loss of generality, we can assume that V is compact. By assumption, .hk /k converges to h uniformly on V and thus, for any ı > 0, there exists k1 2 N such that khk .q/  h.q/k  ı for any k  k1 . However, as h.q/ D x D hk .qk /, we have khk .q/  hk .qk /k  ı. Hence, by Lemma 4, if ı > 0 is sufficiently small, then kq  qk k  M ı. Thus, qk ! q. Therefore, .gk /k is equicontinuous and gk .x/ ! g.x/ for any x 2 . Fix any x 2  and let h.q/ D x. Choose any sufficiently small ı > 0 such that kh.r/  h.q/k  ı or khk .r/  hk .q/k  ı for some k  k0 implies kr  qk  M ı. Then, if ky  xk  ı, kgk .y/k  kh1 k .y/k C 1

1 1  kh1 k .y/  hk .x/k C khk .x/k C 1

 M ı C sup kh1 ` .x/k C 1 < C1; `k0

and thus .gk /k is uniformly bounded on B D fy 2 jky  xk  ıg. If ı > 0 is sufficiently small, then B is compact. Hence, by Ascoli-Arzela theorem, any subsequence of .gk /k has a convergent subsubsequence. Suppose that .gk /k does not converge to g uniformly on B. Then, there exists " > 0 and a subsequence .gm.k/ /k such that sup kgm.k/ .y/  g.y/k  ":

y2B

By the above result, there exists a uniformly convergent subsubsequence .g`.k/ /k . Because gk converges to g pointwise, we conclude that sup kg`.k/ .y/  g.y/k < "

y2B

for some k, a contradiction. This completes the proof.



Acknowledgements We are grateful to Toru Maruyama for his helpful comments and suggestions. We would also like to express gratitude to the anonymous referee for their helpful advice.

A Theory for Estimating Consumer’s Preference from Demand

55

References 1. Afriat S (1967) The construction of a utility function from demand data. Int Econ Rev 8:67–77 2. Antonelli GB (1886) Sulla Teoria Matematica Dell’ Economia Politica. Tipografia del Folchetto, Pisa 3. Debreu G (1972) Smooth preferences. Econometrica 40:603–615 4. Hartman P (1997) Ordinary differential equations. Society for Industrial and Applied Mathematics, Birkhaeuser Verlag AG 5. Hildenbrand W (1974) Core and equilibria of a large economy. Princeton University Press, New Jersey 6. Hosoya Y (2013) Measuring utility from demand. J Math Econ 49:82–96 7. Hurwicz L, Uzawa H (1971) On the integrability of demand functions. In: Chipman JS, Hurwicz L, Richter MK, Sonnenschein HF (eds) Preferences, utility and demand. Harcourt Brace Jovanovich, New York, pp 114–148 8. Kannai Y (1970) Continuity properties of the core of a market. Econometrica 38:791–815 9. Kim T, Richter MK (1986) Nontransitive-nontotal consumer theory. J Econ Theory 34:324– 363 10. Pareto V (1906) Manuale di economia politica con una introduzione alla scienza sociale. Societa Editorice Libraria, Milano 11. Pontryagin LS (1962) Ordinary differential equations. Addison-Wesley (translated from Russian). Reading, Massachusetts 12. Quah JK-H (2006) Weak axiomatic demand theory. Econ Theory 29:677–699 13. Richter MK (1966) Revealed preference theory. Econometrica 34:635–645 14. Samuelson PA (1950) The problem of integrability in utility theory. Economica 17:355–385 15. Smale S, Hirsch MW (1974) Differential equations, dynamical systems, and linear algebra. Academic, New York

Adv. Math. Econ. 19, 57–89 (2015)

Least Square Regression Methods for Bermudan Derivatives and Systems of Functions Shigeo Kusuoka and Yusuke Morimoto

Abstract Least square regression methods are Monte Carlo methods to solve nonlinear problems related to Markov processes and are widely used in practice. In these methods, first we choose a system of functions to approximate value functions. So one of questions on these methods is what kinds of systems of functions one has to take to get a good approximation. In the present paper, we will discuss on this problem. Keywords Computational finance • Option pricing • Malliavin calculus • Least square regression methods

Article Type: Research Article Received: October 1, 2014 Revised: December 1, 2014

JEL Classification: C63, G12. Mathematics Subject Classification (2010): 65C05, 60G40. S. Kusuoka () Graduate School of Mathematical Sciences, The University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8914, Japan e-mail: [email protected] Y. Morimoto Graduate School of Mathematical Sciences, The University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8914, Japan Bank of Tokyo Mitsubishi UFJ, 2-7-1, Marunouchi, Chiyoda-ku, Tokyo e-mail: [email protected] 57

58

S. Kusuoka and Y. Morimoto

1 Introduction Least square regression methods are Monte Carlo methods to solve non-linear problems related to Markov processes. These methods were introduced by LongstaffSchwartz [8] and Tsitsiklis-Van Roy[9] and are widely used in practice. There are many works related to this methods. Concerning the applications for pricing Bermudan derivatives, the convergence to a real price was proved by ClementLamberton-Protter [4] and rate of convergence was studied by Belomestny [2]. In these methods, first we choose a system of functions to approximate value functions. So one of questions on these methods is what kinds of systems of functions one has to take to get a good approximation. In the present paper, we will discuss on this problem. Related topics have been discussed by Gobet-Lemor-Warin [5] and BallyPagés [1]. Let .; F ; P / be a probability space, M = 1; and fGm gM mD0 be a filtration on .; F ; P /: Let .E; B/ a measurable space and m.E/ be the set of Borel measurable functions on E: Let pm W E  B ! Œ0; 1; m D 0; : : : ; M  1; be such that pm .x;
/ W B ! Œ0; 1 is a probability measure on E for any x 2 E; and pm .
; A/ W E ! Œ0; 1 is B-measurable for any A 2 B: Let x0 2 E and fix it throughout. Let X W f0; 1; : : : ; M g   ! E be an E-valued process such that X0 D x0 ; Xm W  ! E is Gm -measurable, m D 0; : : : ; M; and P .XmC1 2 AjGm / D pm .Xm ; A/ a:s:

A 2 B; m D 0; : : : ; M  1:

So X is a Markov process starting from x0 whose transition probability is given by pm .x; dy/: Let m ; m D 1; : : : ; M; be the probability law of Xm ; m D 0; 1; : : : ; M: Then 0 is the probability measure concentrated in x0 ; and mC1 .A/ D

Z

pm .x; A/m .dx/; E

y 2 E; m D 0; 1; : : : ; M  1:

Let Pm W L2 .EI dmC1 / ! L2 .EI dm /; m D 0; 1; : : : ; M  1; be a linear operator given by .Pm f /.x/ D

Z

pm .x; dy/f .y/; E

f 2 L2 .EI dmC1 /:

Now let fm 2 L4 .EI dm /; m D 1; 2; : : : ; M: We define fQm ; fQm 2 L4 .EI dm /; m D 0; 1; 2; : : : ; M; inductively by the following. fQM D fM ;

Least Square Regression Methods

59

and fQm D fQm _ fm ;

fQm1 D Pm .fQm _ fm /;

m D M; M  1; : : : ; 1:

Then it is well-known that fQ0 D supfEŒf .X /I  is a fGm gM mD0 -stopping time with  2 f1; 2; : : : ; M g a:s:g: fQ0 is the price of a Bermudan derivative for which exercisable times are 1; : : : ; M; and pay-off at each time is fm .Xm /; m D 1; : : : ; M: Our concern is to compute fQ0 numerically. Let V denote the set of finite dimensional vector subspaces of m.E/: For any probability measure  on .E; B/; let V ./ denote the subset of V such that V 2 V ./; if and only if V satisfies the following two conditions. R 1. If g 2 V; then E g.x/4 .dx/ < 1: 2. If g 2 V and g.x/ D 0   a:e:x; then g  0: For any probability measure  on .E; B/ and V 2 V ./; we define 0 .V; / and 1 .V; / by the following. R g.x/4 .dx/ 0 .V; / D supf R E I g 2 V n f0gg . E g.x/2 .dx//2 Z dim XV V er .x/2 /2 .dx/I fer gdim 1 .V I / D inff . rD1 is an orthonormal basis E

rD1

of V as a subspace of L2 .EI d/ g:

We will show in Proposition 3 that 1 .V I / 5 .dim V /2 0 .V I / and 0 .V I / 5 1 .V I /: .`/

.`/

.`/

Now let .X0 ; X1 ; : : : ; XM /; ` D 1; 2; : : : ; be independent identically dis` tributed E M C1 -valued random variables such that the law of .X0` ; X1` ; : : : ; XM /; ` D 1; 2; : : : ; is the same as the law of .X0 ; X1 ; : : : ; XM / under P: .L/ For any m D 0; 1; : : : ; M 1; and L = 1; we define Dm W m.E/m.E/ ! Œ0; 1/ by L

.L/ .g; f /.!/ D . Dm .k/

1 X .`/ .g.Xm.`/ .!/  f .XmC1 .!//2 /1=2 ; L `D1

g; f 2 m.E/:

Let Vm ; k D 1; 2; : : : ; be a sequence of strictly increasing vector spaces in S .k/ 2 V .m / such that 1 kD1 Vm is dense in L .EI dm / for m D 1; : : : ; M  1:

60

S. Kusuoka and Y. Morimoto .L/

.L/

Now we assume that gm W  ! Vm ; m D 0; 1; : : : ; M  1; L D 1; 2; : : : ; satisfy the following. .L/

.L/ Dm1 .gm1 .!/; gm .!/ _ fm /.!/ .L/ D inffDm1 .h; gm .!/ _ fm /I h 2 Vm.L/ .!/g

(1)

.L/

for m D 1; 2; : : : ; M: Here we let gM D fM : .L/ We will show that such gm ’s always exist. Then we will prove the following. .L/

Theorem 1. Suppose that 1 .Vm I m /=L ! 0; as L ! 1 for m D 1; : : : ; M 1: Then there are L 2 F ; L D 1; 2; : : : ; and random variables ZL ; L D 1; 2; : : : ; such that P .L / ! 1; as L ! 1; .L/ jfQ0  g0 .!/j 5 ZL .!/;

L = 1; ! 2 L ;

and EŒZL2 ; L 1=2 ! 0; as L ! 1: Moreover, we have EŒZL2 ; L 1=2 56

M 1 X mD1

C5

1  jjL4 .EIdm / 1 .Vm.L/ ; m /1=4 .1 C 0 .Vm.L/ ; m //1=4 jjPm fQmC1 L1=2

M 1 X mD1

  jjL2 .EIdm / :  m;V .L/ Pm fQmC1 jjPm fQmC1 m

.L/

Here m;V .L/ is the orthogonal projection in L2 .E; dm / onto Vm ; m D 1; : : : ; M: m

.L/

So roughly speaking, g0 ! f0 in probability as L ! 1 in a certain rate. It is obvious that 0 .V I m / = 1 and 1 .V I m / = dim V for any V 2 Vm ; m D 1; 2; : : : ; M: So the above theorem raises the following question. Can one   jjL2 .EIdm / for V 2 V .m /? If we  m;V Pm fQmC1 estimate 0 .V I / and jjPm fQmC1 .k/ can do it, we may find a sequence Vm 2 V .m / such that the convergence rate is good. We give an estimate when an underlying process is a 1-dimensional Brownian motion and V is a space of polynomials in Sect. 6. Also, we introduce a random systems of piece-wise polynomials in Sect. 8, and we give some estimates when an

Least Square Regression Methods

61

underlying process is a Hörmander type diffusion process as discussed in [6]. As far as we judge from these estimates, a usual polynomial system is not good, and such a random system of piece-wise polynomials is better.

2 Preliminary Results Let Pf .E  E/ be the set of probability measures on .E  E; B  B/ whose supports are finite subsets of E  E: Let i W E  E; i D 1; 2; be natural projections given by 1 .x; y/ D x; 2 .x; y/ D y; x; y 2 E: For any  2 Pf .E  E/; let S.
; I / W m.E/  m.E/ ! R be given by S.g; f I / D

Z

EE

.g.x/  f .y//2 .dx; dy/;

g; f 2 m.E/:

(2)

Then we have the following. Proposition 1. Let  2 Pf .E  E/: For any f 2 m.E/ and V 2 V ; let s .f I V; / D inffS.g; f I /I g 2 V g and €.f I V; / D fg 2 V I S.g; f I / D s .f; V; /g: Then we have the following. (1) €.f I V; / is not empty for any f 2 m.E/ and V 2 V : (2) Let V 2 V : If f 2 m.E/ and g 2 €.f I V; /; then Z

EE

h.x/.f .y/  g.x//.dx; dy/ D 0 for any h 2 V:

Moreover, if f1 ; f2 2 m.E/; gi 2 €.fi I V; /; i D 1; 2; then S.g1  g2 ; 0I / 5 S.0; f1  f2 I /: (3) If f 2 m.E/; g 2 €.f I V; / and gQ 2 V; then S.g  g; Q 0I /1=2 Z D supfj h.x/.f .y/  g.x//.dx; Q dy/jI h 2 V; S.h; 0I / D 1g: EE

62

S. Kusuoka and Y. Morimoto

Proof. (1) It is easy to see that S.g; f I / = S.0; f I / C S.g; 0I /  2S.g; 0I /1=2 S.0; f I /1=2 ;

g 2 V:

Let V0 D fg 2 V I S.g; 0I / D 0g D fg 2 V W g.x/ D 0 for  -a.e. .x; y/ 2 E  Eg: Then it is easy to see that V0 is a vector subspace of V: So there is a vector subspace V1 of V such that V0 C V1 D V and V0 \ V1 D f0g: It is easy to see that g 2 V1 ! S.g; f I A/ is a continuous function from V1 to Œ0; 1/ and that S.g; f I A/ ! 1 as g ! 1 in V1 : So we see that there is a minimum point g0 2 V1 : Note that S.g C h; f I / D S.g; f I / for any g 2 V and h 2 V0 : Therefore we see that S.g0 ; f I / D s .f I V; / and that €.f I V; / is not empty. (2) Let g 2 €.f I V; /: The first assertion is obvious, since 0D

d S.g C th; f I /jt D0 D dt

Z

EE

h.x/.f .y/  g.x//.dx; dy/

for any h 2 V: Let fi 2 m.E/; gi 2 €.fi I V; /; i D 1; 2: Then we have S.g1  g2 ; f1  f2 I / D S.g1  g2 ; 0I / C S.0; f1  f2 I / Z .g1 .x/  g2 .x//.f1 .y/  g1 .x/  .f2 .y/  g2 .x///.dx; dy/: 2 EE

By the first assertion, we see that S.0; f1  f2 I / D S.g1  g2 ; f1  f2 I / C S.g1  g2 ; 0I /: So we have the second assertion. (3) Let g 2 €.f I V; / and gQ 2 V: Then we have S.gQ C h; f I / D S.g; Q f I / C S.h; 0I /  2

Z

EE

h.x/.f .y/  g.x//.dx; Q dy/:

Let c D supf

Z

EE

h.x/.f .y/  g.x//.dx; Q dy/I h 2 V; S.h; 0I / D 1g = 0:

Then we see that s .f I V; / D S.g; Q f I / C inf.t 2  2tc/ D S.g; Q f I /  c 2 : t =0

Least Square Regression Methods

63

Also, we have by Assertion (2) S.g; Q f; / D S.g C .gQ  g/; f I / D S.g; f I / C S.gQ  g; 0 W / D s .f I A; V / C S.gQ  g; 0 W /: So we see that c 2 D S.gQ  g; 0 W /: This implies our assertion.



For any m D 1; 2; : : : ; M; V 2 V .m /; and  2 Pf .E  E/; let Z h.x/2 m .dx/ D 1g: ım .V I / D supfjS.h; 0I /  1jI h 2 V; E

Then we have the following. Proposition 2. Let m D 1; 2; : : : ; M; V 2 V .m /; and  2 Pf .E  E/: Let fek I k D 1; : : : ; dim V g be an orthonormal basis of V: Here we regard V as a Hilbert subspace of L2 .E; B.E/; dm /; and so we have Z

E

ei .x/ej .x/m .dx/ D ıij ;

i; j D 1; : : : ; dim V:

Let A be a .dim V /  .dim V /-symmetric matrix valued function defined in E given by V dimV A.x/ D .Aij .x//dim i;j D1 D .ei .x/ej .x//i;j D1 ;

x 2 E:

Then ım .V I / is equal to the operator norm of the dim V  dim V -symmetric V matrix AN  I: Here I is the identity matrix and AN D .ANij /dim i;j D1 ; where ANij D

Z

ei .x/ej .x/.dx; dy/; E

i; j D 1; : : : ; dim V :

In particular, ım .V I /2 5

dim XV

.

i;j D1

Z

E

.ei .x/ej .x/  ıij /.dx; dy//2 :

Proof. It is easy to see that ım .V I / D supfjS. D supfj

dim XV

i;j D1

dim XV i D1

ai ei ; 0I /  1jI

ai aj .ANij  ıij /jI

Since AN  I is symmetric, we see our assertion.

dim XV i D1

dimV X i D1

ai2 D 1g

ai2 D 1g: 

64

S. Kusuoka and Y. Morimoto

Proposition 3. For any probability measure  on .E; B/; and V 2 V ./; 1 .V; / 5 .dim V /2 0 .V; / and 0 .V; / 5 1 .V; /: V Proof. Let fer gdim rD1 be an orthonormal basis of V: Then we see that

Z

. E

dim XV

Z

er .x/2 /2 .dx/ 5

.dim V /.

E

rD1

dim XV

er .x/4 /.dx/ 5 .dim V /2 0 .V; /:

rD1

So we have the first assertion. Let g 2 V: Then we have Z

E

g.x/4 .dx/ D 5

Z

.

E

dim XV

Z

.

E

dim XV

.g; er /L2 .d/ er .x//4 .dx/

rD1

.g; er /2L2 .d/ /2 .

rD1

dim XV

er .x/2 /2 .dx/:

rD1

Note that dim XV

.g; er /2L2 .d/

rD1

D

Z

g.x/2 .dx/:

E

So we have the second assertion.



3 Random Measures .L/

For m D 1; : : : ; M; and L = 1; let m be a random probability measure belonging to Pf .E  E/ given by .L/ .A/ D m

1 .`/ #f` 2 f1; : : : ; LgI .Xm1 ; Xm.`/ / 2 Ag; L .L/

A 2 B  B:

For any m D 0; 1; : : : ; M  1; and L = 1; we define Nm W m.E/   ! Œ0; 1/ by L

Nm.L/ .f

1 X f .Xm.`/ .!//2 /1=2 : /.!/ D . L `D1

Least Square Regression Methods

65

Then we see that .L/

.L/ /; Nm1 .g/ D S.g; 0I m

g 2 m.E/; m D 1; : : : ; M:

Then we have the following. Proposition 4. Let m D 1; : : : ; M  1; L = 1; and V 2 V .m /: Then we have the following. .L/

(1) If ım .V I m / 5 1=2; then 1 .L/ .g/2 5 N 2 m1

Z

.L/

E

g.x/2 m .dx/ 5 2Nm1 .g/2 ;

g 2 V:

(2) .L/ 2 EŒım .V I m / 5

1 1 .V; m /: L

In particular, we have .L/ /> P .ım .V I m

4 1 / 5 1 .V; m /: 2 L

.L/

Proof. (1) Suppose that ım .V I m / 5 1=2: If h 2 V and then from the definition we have

R

E

h.x/2 m .dx/ D 1;

1 .L/ 5 Nm1 .h/2 5 2: 2 So we have our assertion. V (2) Let fer gdim rD1 be an orthonormal basis of V: It is easy to see that .L/ 2 EŒım .V I m / 5

dim XV

L

EŒ.

r;r 0 D1

1X .er .Xm` /er 0 .Xm` /  ır;r 0 //2  L `D1

dim V Z dim V Z 1 X 1 X 2 D .er .x/er 0 .x/  ır;r 0 / m .dx/ 5 er .x/2 er 0 .x/2 m .dx/: L 0 L 0 E E r;r D1

D

1 L

Z

. E

dim XV

r;r D1

er .x/2 /2 m .dx/:

rD1

So we have the first part of our assertion. The second part is an easy consequence of Chebyshev’s inequality. 

66

S. Kusuoka and Y. Morimoto

For any m D 1; 2; : : : ; M 1; and V 2 V .m /; let €O m;V W m.E/Pf .E E/ ! V be defined by the following. g D €O m;V .f; /; f 2 m.E/;  2 Pf .E  E/; if g 2 €.f; V I / and Z

E

g.x/2 m .dx/ D inff

Z

E

g.x/ Q 2 m .dx/I gQ 2 €.f; V I /g:

€O m;V is well-defined by Proposition 1 and the definition of V .m /: Let F W E   ! R be B  F -measurable function. Then it is easy to see that .L/ the mapping ! 2  ! s .F .
; !/; V; m .!// is F -measurable. So we see that the .L/ mapping ! 2  ! €O m;V .F .
; !/; m .!// is also F -measurable (see Castaing [3] for example). For V 2 V .m /; m D 1; : : : ; M; let m;V W L2 .EI dm / ! V be the orthogonal projection onto V: Then we have the following. Proposition 5. Let m D 1; : : : ; M  1; and L = 1: Then for V 2 Vm and f 2 L4 .E; B.E/; dmC1 /; we have 1 .L/ 2 .L/ EŒNm.L/ .m;V Pm f  €O m;V .f I m // ; ım .V; m /5  2 Z 8 5 .1 .V; /.1 C 0 .V; ///1=2 . f .y/4 mC1 .dy//1=2 : L E V Proof. Let g D m;V Pm f; and fer gdim rD1 be an orthonormal basis of V: Note that 1 EŒer .Xm1 /.f .XmC1 /  g.Xm1 // D

D

Z

E

Z

EE

er .x/.f .y/  g.x//m .dx/pm .x; dy/

er .x/.Pm f .x/  g.x//m .dx/ D 0;

r D 1; : : : ; dim V:

By Proposition 1(3) we see that 1 .L/ 2 .L/ EŒNm.L/ .g  €O m;V .f I m // ; ım .V; m /5  2 Z Z .L/ 2 5 2EŒsupfj h.x/.f .y/g.x//mC1 .dx; dy/j I h 2 V; h.x/2 m .dx/D1g E

EE

D 2EŒsupfj

dim XV rD1

ar

Z

EE

.L/

er .x/.f .y/  g.x//mC1 .dx; dy/j2 I

dim XV rD1

ar2 D 1g

Least Square Regression Methods dim XV

D 2EΠD2 D

.

rD1

dim XV

Z

67

.L/

EE

er .x/.f .y/  g.x//mC1 .dx; dy//2 

L

EŒ.

rD1

1 X ` /  g.Xm` ///2  er .Xm` /.f .XmC1 L `D1

dim V 2 X 1 EŒer .Xm1 /2 .f .XmC1 /  g.Xm1 //2  L rD1

dim V Z 2 X er .x/2 .f .y/  g.x//2 m .dx/pm .x; dy/ D L rD1 EE

5

2 . L

Z

E

.

dim XV

er .x/2 /2 m .dx//1=2 .

rD1

Z

EE

.f .y/  g.x//4 m .dx/pm .x; dy//1=2 :

Note that Z Z .f .y/g.x//4 m .dx/pm .x; dy/516 EE

D16.

Z

EE

Z

f .y/4 mC1 .dy/C

E

.f .y/4 Cg.x/4 /m .dx/pm .x; dy/ g.x/4 m .dx//:

E

By Proposition 3, we see that Z

4

g.x/ m .dx/ 5 0 .V; m /. E

5 0 .V; m /

Z

Z

.Pm f /.x/2 m .dx//2

E

f .y/4 mC1 .dy/:

E

So we have our assertion.



The following is obvious. Proposition 6. Let m D 1; : : : ; M; and L L2 .E; B.E/; dm /; we have EŒNm.L/ .f /2  D

Z

=

1: Then for any f

f .x/2 m .dx/: E

2

68

S. Kusuoka and Y. Morimoto

4 Proof of Theorem 1 Now let us think of the setting in Introduction. Let m W E R ! R; m D 1; : : : ; M; be given by m .x; z/ D fm .x/ _ z;

x 2 E; z 2 R; m D 1; 2; : : : ; M:

Then we see that jm .x; z1 /  m .x; z2 /j 5 jz1  z2 j;

x 2 E; z1 ; z2 2 R; m D 1; : : : ; M:

Note that fQm .x/ D m .x; fQm .x// and fQm1 D Pm1 fQm ;

m D 1; : : : ; M:

.L/

.L/

Remind that Vm 2 V .m /; L = 1; m D 1; : : : ; M: Let us take gm W  ! .L/ Vm ; m D M; : : : ; 0; such that .L/

gM .!/ D fM ; .L/

L .L/ gm .!//; .!/ 2 €.mC1 .
; gmC1 .!/.
//; Vm.L/ I m

m D M  1; : : : ; 0:

.L/ Then we see that Eq. (1) is satisfied. Let ZQ m ; m D 0; 1; : : : ; M  1; be given by .L/ ZQ m   L   €O m;Vm;L .fQmC1 I m //: /CNm.L/ .m;V .L/ Pm fQmC1 m;V .L/ Pm fQmC1 D Nm.L/ .Pm fQmC1 m

m

.L/

Also, let Zm ; m D 0; 1; : : : ; M  1; be given by .L/

Z0

D

M 1 X kD0

.L/ ZQ k ;

and .L/ Zm

D jjfQm  m;V .L/ fQm jjL2 .E;dm / C 2Nm .fQm  m;V .L/ fQm ; !/ C 2 m

m

M 1 X kDm

.L/ ZQ k ;

Least Square Regression Methods

69

m D 1; : : : ; M  1: Finally, let L D

M 1 \ mD1

.L/ /5 fım .Vm.L/ I m

1 g: 2

Then we have the following. .L/ .L/ Proposition 7. (1) jfQ0  g0 .!/j 5 Z0 : .L/ (2) For any ! 2  ; .L/ .L/ jjfQm  .gm .!/ _ fm /jjL2 .EIdm / 5 jjfQm  gm .!/jjL2 .EIdm / .L/ 5 Zm ;

m D 1; : : : ; M:

(3) P . n L / 5

M 1 X kD1

4 .L/ 1 .Vk ; k /; L

and .L/ 2 EŒjZm j ; L 1=2

56

M 1 X kD1

C5

M 1 X kD1

1 .L/ .L/  f. 1 .Vk ; k /1=2 .1 C 0 .Vk ; k //1=2 g1=2 jjPk fQkC1 jjL4 .EIdk / L

  jjL2 .EIdk / ;  k;V .L/ Pk fQkC1 jjPk fQkC1 k

m D 0; 1; : : : ; M  1:

Proof. Note that .L/ Nm.L/ .fQm gm .!/; !/

  L  L .L/ €O m;Vm;L .fQmC1 I m /; !/CNm.L/ .€O m;Vm;L .fQmC1 I m //gm .!/; !/: 5Nm.L/ .Pm fQmC1

By Proposition 1(2), we have .L/ L  .!/; !/ //  gm I m Nm.L/ .€O m;Vm;L .fQmC1

5 NmC1 .mC1 .
; fQmC1 .
//  mC1 .
; gmC1 .!/.
//; !/ .L/

.L/

.L/ .L/ 5 NmC1 .fQmC1  gmC1 .!/.
/; !/:

70

S. Kusuoka and Y. Morimoto

So we see that .L/ Nm.L/ .fQm  gm .!/; !/ 5

M 1 X kDm

.L/   I kL /; !/:  €O k;Vk;L .fQkC1 Nk .Pk fQkC1

Then we have .L/ .!/; !/ 5 Nm.L/ .fQm  gm

M 1 X kDm

.L/ ZQ k :

In particular, jfQ0  g0 .!/j 5 .L/

M 1 X kD0

.L/ .L/ ZQ k D Z0 :

This implies Assertion (1). Also, we see that if ! 2 L ; then .L/ .!/jjL2 .E;dm / jjfQm  gm

.L/ .!/jjL2 .E;dm / 5 jjfQm  m;V .L/ fQm jjL2 .E;dm / C jjm;V .L/ fQm  gm m

m

.L/ .!/; !/ 5 jjfQm  m;V .L/ fQm jjL2 .E;dm / C Nm.L/ .m;V .L/ fQm  gm m

m

5 jjfQm  m;V .L/ fQm jjL2 .E;dm / j C 2Nm .fQm  m;V .L/ fQm ; !/ m

m

C2Nm.L/ .fQm



.L/ gm .!/; !/:

This implies Assertion (2). The first assertion of (3) is obvious from Proposition 4. By Propositions 5 and 6, we have .L/ 2 EŒ.ZQ m / ; L 1=2

5 jjfQm  m;V .L/ Pm fQm jjL2 .E;dm / m

1  C3. .1 .V; /.1 C 0 .V; //1=2 /1=2 jjfQmC1 jjL4 .EIdmC1 / : L So we have the second assertion of (3).



Theorem 1 follows from Proposition 7 immediately. The following is an easy consequence of Proposition 7. .L/

Proposition 8. Assume that 1 .Vm I m /=L ! 0; L ! 1; m D 1; : : : ; M  1: Let ı 2 .0; 1/; and let

Least Square Regression Methods

dL D

M 1 X

71

.L/ 2 / ; L 1=2 ; EŒ.Zm

L = 1;

mD0

Q ı 2 F ; L = 1; be given by and let  L Q ıL D L \ 

M 1 \ mD1

.L/ 5 dL1ı g: fZm

Q ı / ! 1; L ! 1: Also, we have Then dL ! 0; and P . L jjfQm  gm .!/jjL2 .EIdm / 5 dL1ı ;

Q ıL ; L = 1: m D 1 : : : ; M; ! 2 

5 Re-simulation Let us be back to the situation in Introduction. Let hm 2 L2 .EI dm /; m D 1; : : : ; M; with hM D fM : Let  a stopping time given by  D minfk D 0; 1; : : : ; M I fk .Xk / = hk .Xk /g; and let M 1 / D EŒf .X /: c0 D c0 .fhm gmD1

Then we have the following. Proposition 9. Let ˇ = 0: Assume that there is a C0 > 0 such that m .fjfm  fQm j 5 "g/ 5 C0 "ˇ ;

" > 0; m D 1; 2; : : : ; M:

Then we have jfQ0  c0 j 5 .C0 C 1/

M 1 X mD1

jjfQm  hm jjL2 .EId

1Cˇ=.2Cˇ/ : m/

Proof. Let hO m ; m D M; M  1; : : : ; 0; be inductively given by hO M D fM D hM ; hO m1 D Pm1 .1ffm =hm g fm C 1ffm 0g < 1: These and Sobolev’s inequality imply that there is a C > 0 such that t N `0 h.x/4N `0 CmC1 p.t; x; y/." C h.y//m 5 C; for any x 2 E; y 2 RN ; t 2 .0; T ; and " > 0: This proves our assertion. Let Pt ; t = 0; be a diffusion operator defined in Cb1 .RN / given by .Pt f /.x/ D EŒf .X.t; x//;

f 2 Cb1 .RN /:

Then we see that .Pt f /.x/ D Then we have the following.

Z

p.t; x; y/f .y/dy; E

x 2 E:



Least Square Regression Methods

81

Proposition 15. For any T > 0 and ˛ 2 ZN =0 ; there is a C 2 .0; 1/ such that j

@˛ .Pt f /.x/j 5 C t .j˛jCN C2/`0 =2 h.x/2.j˛jCN C2/`0 .Pt .jf j2 /.x//1=2 @x ˛

for any t 2 .0; T ; x 2 E and f 2 Cb1 .RN /: Proof. By Proposition 13, we see that there is a C1 2 .0; 1/ such that for any f 2 Cb1 .RN / j

@˛ .Pt f /.x/j 5 @x ˛

Z

@˛ p .t; x; y/jjf .y/jdy ˛ E @x Z .j˛jC1/`0 =2 2.j˛jC1/`0 h.x/ p.t; x; y/2N=.2N C1/ jf .y/jdy 5 C1 t j

E

5 C1 t .j˛jC1/`0 =2 h.x/2.j˛jC1/`0 j j

Z

Z

f .y/2 p.t; x; y/dyj1=2 E

p.t; x; y/.2N 1/=.4N C2/ dyj1=2 : E

By Proposition 12, we see that there is a C2 > 0 such that Z

p.t; x; y/.2N 1/=.4N C2/ dy 5 C2 t .N C1/`0 =4 h.x/.N C1/`0 ; E

x 2 E; t 2 .0; T :

So we have our assertion.



The following is an easy consequence of Proposition 13. Proposition 16. For any ˇ 2 .0; 1=N / and T > 0; there is a C > 0 such that j

@ .p.t; x; y/ˇ /j 5 C t `0 h.x/4`0 ; @y i

x 2 E; t 2 .0; T :

8 A Random System of Piece-Wise Polynomials Let  be a probability measure on RN : For any m = 2; let .m/ DE k

N Y .2ki  m/ .2.ki  1/  m/ log m; log m/; D Πm m i D1

.m/ for kE D .k1 ; : : : ; kN / 2 f1; : : : ; mgN : Let Dm D fD E I kE 2 f1; : : : ; mgN g: Then k S we have Dm D Πlog m; log m/N :

82

S. Kusuoka and Y. Morimoto

Let X1 ; X2 ; : : : ; i.i.d. random variables defined on a probability space .; F ; P / whose distributions are : Let Dm;n .!/; m; n =; ! 2 ; be a random sub-family of Dm given by Dm;n .!/ D fD 2 Dm I there is a k 2 f1; : : : ; ng such that Xk .!/ 2 D g: Let Pr ; r D 0; 1; 2; : : : ; be the set of polynomials on RN of degree less than or equal to r: Now let Vn;m;r .!/; m; n = 2; r = 0; ! 2 ; be a finite dimensional vector subspace of m.RN / hulled by f 1D ; f 2 Pr ; D 2 Dm;n .!/: It is obvious that dim Vn;m;r .!/ 5 N m .N C 1/r : Now let us use the notation in the previous section. Let X.t; x/; t 2 Œ0; 1/; x 2 RN ; be the solution to the SDE (3) and we assume the (UFG) condition holds. Let x0 2 RN such that h.x0 / > 0; and so x0 2 E: Let T0 > 0 and .x/ D p.T0 ; x0 ; x/; x 2 RN : We think of the case that .dx/ D .x/dx: Then we have the following. Theorem 2. Let r = 0; ı > 0; > 0; and T > 0; and let nm ; m D; 2; : : : ; be integers satisfying mN C 5 nm < 2mN C : Then there are m 2 F ; m D 1; 2; : : : ; and C 2 .0; 1/ satisfying the following. (1) P .m / ! 1; m ! 1: (2) For any ! 2 m ; inf .x/ =

x2D

1 sup .x/; 2 x2D

and .D/ = C 1 m.2N C Cı/ for any D 2 Dm;nm .!/ and m = 2: (3) For any ! 2 m ; 0 .Vm;nm ;r ; / 5 C m2N C Cı : (4) For any ! 2 m ; f 2 Cb1 .RN / and t 2 .0; T ; jjPt f  Vm;nm ;r Pt f jjL2 .d/ Z 5 C.t .rC2N C3/`0 m.rC1/Cı C m =4Cı /. f .y/4 p.T0 C t; x0 ; y/dy/1=4 : RN

Here Vm;n;r is the orthogonal projection in L2 .EI d/ onto Vm;n;r .!/: We make some preparations to prove Theorem 2. Proposition 17. For any r = 0; there is a Cr > 0 such that .

Z

.";"/N

f .y/4 dy/1=4 5 Cr "N=4 .

for any " > 0 and f 2 Pr :

Z

.";"/N

f .y/2 dy/1=2

Least Square Regression Methods

83

Proof. Let us fix n = 0: Since Pr is a finite dimensional vector space, any norms on Pr are equivalent. So we see that there is a Cr > 0 such that .

Z

.1;1/N

jf .x/j4 dx/1=4 5 Cr .

Then we see that Z .

.";"/N

5 Cr "

N=4

.

Z

Z

.1;1/N

f .x/4 dx/1=4 D "N=4 . 2

f ."x/ dx/ .1;1/N

1=2

jf .x/j2 dx/1=2 ;

Z

f ."x/4 dx/1=4

.1;1/N

D Cr "

f 2 Pr :

N=4

.

Z

f .x/2 dx/1=2 : .";"/N

This implies our assertion.

 Pn

For any Borel subset A in RN and n; let Nn .A/ be Nn .A/ D kD1 1A .Xi /: Let > 0 and ı 2 .0; =2/; and fix them. Let 0 D N C  ı=3 and 1 D .0/ .1/ 2N C C ı=3: Now let Dm and Dm be subsets of Dm ; m = 1; given by Dm.0/ D fD 2 Dm I .D/ = m 0 g; and Dm.1/ D fD 2 Dm I .D/ = m 1 g: .0/

.1/

Then it is obvious that Dm  Dm : Then we have the following. Proposition 18. (1) Let 0;m;n ; m = 2; n = 1; be the set of ! 2  such that .0/ Dm  Dm;n .!/: Then we have P . n 0;m;n / 5 mN exp.nm.N C / mı=3 /;

n = 1; m = 2: .1/

(2) Let 1;m;n ; m = 2; n = 1; be the set of ! 2  such that Dm;n .!/  Dm : Then there is an m1 = 1 such that P . n 1;m;n / 5 .2 log 2/nm.N C / mı=3

n = 1; m = m1 :

Proof. Since .1  1=x/x ; x 2 .1; 1/ is increasing in x; we see that 1 1 5 .1  /x 5 e 1 ; 4 x

x = 2:

For D 2 Dm we have P .Nn .D/ D 0/ D .1  .D//n D ..1  .D//1=.D/ /n.D/ :

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Thus we see that P .Nn .D/ D 0/ 5 exp.n.D// for any D 2 Dm ; and 22n.D/ 5 P .Nn .D/ D 0/ for any D 2 Dm with .D/ 2 Œ0; 1=2: So we see that for any D 2 Dm with .D/ 2 Œ0; 1=2; P .Nn .D/ = 1/ 5 1  exp..2 log 2/n.D// 5 .2 log 2/n.D/: Note that .D/ 5 .2m1 log m/N sup .x/: x2RN

So there is an m1 = 1 such that .D/ 5 1=2 for D 2 Dm ; m = m1 : Therefore we see that X P .Nn .D/ D 0/ 5 mN exp.nm 0 /; P . n 0;m;n / 5 .0/

D2Dm

and P . n 1;m;n / 5

X

P .Nn .D/ = 1/ 5 .2 log 2/nmN  1

m = m1 :

.0/ D2Dm nDm

So we have our assertions.



Proposition 19. There is an m2 = 1 satisfying the following. .1/ If D 2 Dm ; then inf .x/ =

x2D

1 sup .x/ = m.N C C2ı=3/ ; 2 x2D

m = m2 :

.1/ N Proof. Assume that D 2 Dm : Let x1 2 DN be a maximal point of .x/; x 2 D: Then we see that .x1 / = .2 log m/N mN  ı=3 : Applying Proposition 16 for ˇ D 1=.2.N C C ı=3// > 0; we see that there is a C0 > 0 such that

j.x/ˇ  .y/ˇ j 5 C0 jx  yj;

x; y 2 RN :

Least Square Regression Methods

85

So we see that j.x/ˇ  .x1 /ˇ j 5 C0

2N log m : m

x 2 D;

and so .x/ˇ = .x1 /ˇ  C0

2N log m m

1 2N log m = . .x1 //ˇ C .1  2ˇ /.2 log m/Nˇ m1=2  C0 2 m So we see that if m is sufficiently large inf .x/ =

x2D

1 sup .x/ = m.N C C2ı=3/ : 2 x2D

Thus we have our assertion.



Proposition 20. There is an m3 = 1 satisfying the following. If ! 2 1;n;m and m = m3 ; then 0 .Vm;n;r .!/I / 5 m2N C Cı : Proof. Let m2 = 1 be as in Proposition 19. Suppose that ! 2 1;n;m and m = m2 : .1/ Then Dm;n .!/  Dm : Let f 2 Vm;n;r .!/: Then there are fD 2 Pr ; D 2 Dm:n .!/; such that f D Then we see that Z f .x/4 .dx/ D RN

X

Z

X

fD 1D :

D2Dm:n .!/

4

fD .x/ .dx/ 5

D2Dm:n .!/ D

52

X

D2Dm:n .!/

52

X

D2Dm:n .!/

inf

x2D

X

sup .x/

D2Dm:n .!/

.x/Cr4 .2m1 1

infx2D .x/

log m/

N

.

Z

Z

fD .x/4 dx D

fD .x/2 dx/2 D

Cr4 .2m1 log m/N .

5 m2N C Cı .2N C1 Cr4 mı=3 .log m/N /. This implies our assertion.

x2D

Z

Z

fD .x/2 .dx//2 D

f .x/2 .dx//2 :

RN



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Proposition 21. For any r = 0; there is a C 2 .0; 1/ satisfying the following. inff.

Z

.";"/N

jf .x/  g.x/j2 dx/1=2 I g 2 Pr g X

5 C "rC1

.

˛2ZN =0 ;rC15j˛j5rCN C1

Z

j

.";"/N

@˛ f .x/j2 dx/1=2 @x ˛

for any f 2 C 1 .RN / and " 2 .0; 1: Proof. By Sobolev’s inequality, we see that there is a C0 > such that sup x2.1;1/N

jf .x/j 5 C0

X

.

˛2ZN =0 ;j˛j5N

Z

j

.1;1/N

@˛ f .x/j2 dx/1=2 ; @x ˛

f 2 C 1 .RN /:

So we see that sup x2.";"/N

jf .x/j 5 C0 X

5 C0

X

˛2ZN =0 ;j˛j5N

"

j˛jN=2

˛2ZN =0 ;j˛j5N

.

j

Z

Z

.1;1/N

.";"/N

For any f 2 C 1 .RN /; jf .x/ 

X

˛2ZN =0 ;j˛j5r

Z

1 @˛ f .0/x ˛ j 5 ˛Š @x ˛

5 jxjrC1

X

˛2ZN =0 ;j˛jDrC1

j

t

j

@˛ .f ."x//j2 dx/1=2 @x ˛

@˛ f .x/j2 dx/1=2 : @x ˛

.1  t/r d rC1 j rC1 f .tx/jdt rŠ dt

0

sup j

t 2Œ0;1

@˛ f .tx/j; @x ˛

and so we have Z inff. jf .x/  g.x/j2 dx/1=2 I g 2 Pr g .";"/N

5 .2N "/rC1CN=2

X

˛2ZN =0 ;j˛jDrC1

5 "rC1 C0 .2N /rC1CN=2

sup x2.";"/N

X

j

@˛ f .x/j @x ˛

˛;ˇ2ZN =0 ;j˛jDrC1;jˇj5N

This implies our assertion

.

Z

..";"/N

j

@˛Cˇ f .x/j2 dx/1=2 : @x ˛Cˇ 

Least Square Regression Methods

87 .1/

Proposition 22. For any T > 0 there is an m4 = 1 such that for any D 2 Dm ; m = m4 ; Z inff jPt f .x/  g.x/j2 .dx/I g 2 Pr g D

5 m2.rC1/C2ı=3 t .rC2N C3/`0 =2

Z

D

Pt .jf j2 /.x/.dx/;

t 2 .0; T ; f 2 Cb1 .RN /: Proof. Let m2 = 1 be as in Proposition 19. Then .x/ = m.N C C2ı=3/ ;

x 2 D; D 2 Dm.1/

for any m = m2 : By Proposition 14, there is a C0 > 0 such that h.x/ = C0 mı=.8.rC2N C3/`0 / ;

x 2 D; D 2 Dm.1/ ; m = m2 :

Then by Proposition 15 we see that there is a C1 > 0 such that X

˛2ZN =0 ;rC15j˛j5N CrC1

j

@˛ Pt f .x/j 5 C1 mı=4 t .rC2N C3/`0 =2 .Pt .jf j2 /.x//1=2 ; @x ˛

.1/

for any x 2 D; D 2 Dm ; m = m2 ; and f 2 Cb1 .RN /: Then by Proposition 21 we .1/ see that there is a C2 > 0 such that for D 2 Dm ; m = m2 ; Z inff. jPt f .x/  g.x/j2 .dx//1=2 I g 2 Pr g D

5 .sup .x// x2D

1=2

inff.

Z

D

jPt f .x/  g.x/j2 dx/1=2 I g 2 Pr g

5 2. inf .x//1=2 C2 .2m1 log m/rC1 x2D

X

.

˛2ZN =0 ;rC15j˛j5rCN

5 2C2 .2m1 log m/rC1 C1 mı=4 t .rC2N C3/`0 =2 .

Z

D

Z

D

j

@˛ Pt f .x/j2 dx/1=2 @x ˛

.Pt .jf j2 //.x/.dx//1=2 :

So we have our assertion. Proposition 23. Let A0;m D

 S

.0/ Dm :

Then there is an m5 = 1 such that

.RN n A0;m / 5 m Cı ;

m = m5 :

88

S. Kusuoka and Y. Morimoto

Proof. We see by Proposition 12 that .RN n A0;m / D .Πlog m; log m/N n A0;m / C .RN n Πlog m; log m/N / Z X D p.T0 ; x0 ; x/dx .D/ C RN nΠlog m;log m/N /

.0/

D2Dm nDm

5m

N  0

C

.N C1/`0 =2 C T0 h.x0 /2.N C1/`0

Z

exp.

RN nΠlog m;log m/N /

2ı0 jx  x0 j2 /dx: T0

This implies our assertion.



Proposition 24. Let r = 0; and T > 0: There is an m6 = 2 satisfying the following. For any ! 2 0;m;n ; m = m6 ; n = 1; jjPt f  Vm;n;r Pt f jjL2 .d/ Z .rC2N C3/`0 =2 .rC1/Cı=2  =4Cı=2 f .y/4 p.T0 C t; x0 ; y/dy/1=4 5 .t m Cm /. RN

for any t 2 .0; T ; and f 2 Cb1 .RN /: Proof. Let m4 ; m5 = 2 be as in Propositions 22 and 23. Let ! 2 0;m;n ; and m = .0/ m4 _ m5 : Then we see that Dm;n .!/  Dm and so we see that inff D

X

Z

RN

inff

D2Dm;n .!/

X

m

Z

D

C 5

jPt f .x/  g.x/j2 .dx/I g 2 Pr g

Z

j.Pt f /.x/  g.x/j2 .dx/I g 2 Pr g

S RN n Dm;n .!/

jPt f .x/j2 .dx/

2.rC1/C2ı=3 .rC2N C3/`0

t

D2Dm;n .!/ N

C.R n A0;m /

1=2

.

Z

RN

5 m2.rC1/C2ı=3 t .rC2N C3/`0 Cm. ı/=2 .

Z

Z

D

Pt .jf j2 /.x/.dx/

jPt f .x/j4 .dx//1=2

RN

RN

Z

f .y/2 p.T0 C t; x0 ; y/dy

f .y/4 p.T0 C t; x0 ; y/dy/1=2 :

So this and Proposition 23 imply our assertion.



Now we have Theorem 2 from Propositions 18, 19, 20 and 24, letting m D 0;m;nm \ 1;m;nm :

Least Square Regression Methods

89

References 1. Bally V, Pagés G (2003) A quantization algorithm for solving multi-dimensional discrete-time optimal stopping problems. Bernoulli 9:1003–1049 2. Belomestny D (2011) Pricing Bermudan options by nonparametric regression: optimal rates of convergence for lower estimates. Financ Stoch 15:655–683 3. Castaing C, Valadier M (1977) Convex analysis and measurable multifunctions. Lecture notes in mathematics, vol 580. Springer, Berlin/New York 4. Clement E, Lamberton D, Protter P (2002) An analysis of a least squares regression algorithm for American option pricing. Financ Stoch 6:449–471 5. Gobet E, Lemor J-P, Warin X (2005) A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann Appl Probab 15:2172–2202 6. Kusuoka S, Morimoto Y (2014) Stochastic mesh methods for Hörmander type diffusion processes. In: Kusuoka S, Maruyama T (eds) Advances in mathematical economics, vol 18. Springer, Tokyo Heidelberg New York Dordrecht London, pp 61–99 7. Kusuoka S, Stroock DW (1985) Applications of Malliavin calculus II. J Fac Sci Univ Tokyo Sect IA Math 32:1–76 8. Longstaff F, Schwartz E (2001) Valuing American options by simulation: a simple least-squares approach. Rev Financ Stud 14:113–147 9. Tsitsiklis J, Van Roy B (1999) Regression methods for pricing complex American style options. IEEE Trans Neural Netw 12:694–703

Adv. Math. Econ. 19, 91–135 (2015)

Discrete Time Optimal Control Problems on Large Intervals Alexander J. Zaslavski

Abstract In this paper we study the structure of approximate solutions of an autonomous nonconcave discrete-time optimal control system with a compact metric space of states. In the first part of the paper we discuss our recent results for systems described by a bounded upper semicontinuous objective function which determines an optimality criterion. These optimal control systems are discretetime analogs of Lagrange problems in the calculus of variations. It is known that approximate solutions are determined mainly by the objective function, and are essentially independent of the choice of time interval and data, except in regions close to the endpoints of the time interval. Our goal is to study the structure of approximate solutions in regions close to the endpoints of the time intervals. The second part of the paper contains new results on the structure of solutions of optimal control systems which are discrete-time analogs of Bolza problems in the calculus of variations. These systems are described by a pair of objective functions which determines an optimality criterion. Keywords Good program • Optimal control problem • Overtaking optimal program • Turnpike property

JEL Classification: C02, C61, C67 Mathematics Subject Classification (2010): 49J99 A.J. Zaslavski () Department of Mathematics, The Technion – Israel Institute of Technology, Technion City, Haifa 32000, Israel e-mail: [email protected] 91

92

A.J. Zaslavski

Article type: Research Article Received: October 1, 2014 Revised: November 7, 2014

1 Introduction The growing significance of the study of (approximate) solutions of variational and optimal control problems defined on infinite intervals and on sufficiently large intervals has been further realized in the recent years [2, 4–11, 16, 18, 21, 22, 24, 25, 27, 37, 38]. This is due not only to theoretical achievements in this area, but also because of numerous applications to engineering [1, 14, 27], models of economic dynamics [13, 17, 20, 23, 26–28, 37, 38], the game theory [12, 27, 34], models of solid-state physics [3] and the theory of thermodynamical equilibrium for materials [15, 19]. In the present paper we analyze the structure of approximate solutions of a discrete-time optimal control system on large intervals, describing a general model of economic dynamics [28–33, 36, 37]. More precisely, we study the structure of approximate solutions of an autonomous nonconcave discrete-time optimal control system with a compact metric space of states. In the first part of the paper we discuss our recent results for systems described by a bounded upper semicontinuous objective function which determines an optimality criterion. These optimal control systems are discrete-time analogs of Lagrange problems in the calculus of variations. The second part of the paper contains new results on the structure of solutions of optimal control systems which are discrete-time analogs of Bolza problems in the calculus of variations. These systems are described by a pair of objective functions which determines an optimality criterion. Let .X; / be a compact metric space and  be a nonempty closed subset of X  X. A sequence fxt g1 t D0  X is called an ./-program if .xt ; xt C1 / 2  for all 2  X , where integers T1 , T2 satisfy 0  T1 < integers t  0. A sequence fxt gTt DT 1 T2 , is called an ./-program if .xt ; xt C1 / 2  for all integers t 2 ŒT1 ; T2  1. We study the problem T 1 X t D0

1 v.xt ; xt C1 / ! max; f.xt ; xt C1 /gtTD0  ; x0 D z1 ; xT D z2

(P1)

considered in [29, 37], the problem T 1 X t D0

1  ; x0 D z1 v.xt ; xt C1 / ! max; f.xt ; xt C1 /gtTD0

(P2)

Discrete Time Optimal Control Problems on Large Intervals

93

considered in [28, 37] and the problem T 1 X t D0

1  ; v.xt ; xt C1 / ! max; f.xt ; xt C1 /gtTD0

(P3)

which was studied in [36], where T is a natural number, z1 ; z2 2 X and v W  ! R1 is a bounded upper semicontinuous objective function. These optimal control problems are discrete-time analogs of Lagrange problems in the calculus of variations. In models of economic growth the set X is the space of states, v is a utility function and v.xt ; xt C1 / evaluates consumption at moment t. The interest in discrete-time optimal problems of types (P1)–(P3) also stems from the study of various optimization problems which can be reduced to it [3, 13–15, 19, 27]. Optimization problems of the types (P1)–(P3) with  D X  X were considered in [27]. In [28–33, 36, 37] we analyzed a turnpike phenomenon for the approximate solutions of problems (P1)–(P3) which is independent of the length of the interval T , for all sufficiently large intervals. The turnpike phenomenon holds if the approximate solutions of the optimal control problems are determined mainly by the objective function v, and are essentially independent of T and z1 ; z2 . Turnpike properties are well known in mathematical economics. The term was first coined by Samuelson in 1948 (see [26]). Problems (P1)–(P3) were analyzed in [28, 29, 36, 37], where we showed, under certain assumptions, that the turnpike property holds and that the turnpike xN is a unique maximizer of the optimization problem v.x; x/ ! max, .x; x/ 2 . Namely, we considered a collections of .v/-good programs which are approximate solutions of the corresponding infinite horizon optimal control problem associated with the objective function v. It was shown that the turnpike property holds and xN is the turnpike if the following asymptotic turnpike property holds: the all .v/-good programs converge to x. N It [36] we showed that the asymptotic turnpike property holds for most cost functions in the sense of Baire category. In other words, the asymptotic turnpike property holds for a generic (typical) cost function. In the first part of this paper we discuss the structure of approximate solutions of the problems (P2) and (P3) in regions close to the endpoints of the time intervals and present the results obtained in [39]. The results of [39] show that in regions close to the right endpoint T of the time interval approximate solutions are determined only by the objective function, and are essentially independent of the choice of interval and endpoint value z1 . For the problems (P3), approximate solutions are determined only by the objective function also in regions close to the left endpoint 0 of the time interval. N D f.y; x/ 2 X  X W .x; y/ 2 g and v.y; More precisely, we define  N x/ D v.x; y/ for all .x; y/ 2  and consider the collection P.v/ N of all solutions of a corresponding infinite horizon optimal control problem associated with the pair N For given  > 0 and an integer   1, we show that if T is large enough and .v; N /.

94

A.J. Zaslavski

fxt gTtD0 is an approximate solution of the problem (P2), then .xT t ; yt /   for all integers t 2 Œ0; , where fyt g1 N t D0 2 P.v/. Moreover, using the Baire category approach, we show that for most objective functions v the set P.v/ N is a singleton. In the second part of the paper we establish new results on the structure of solutions of optimal control systems which are discrete-time analogs of Bolza problems in the calculus of variations. These systems are described by a pair of objective functions which determines an optimality criterion. Our paper is organized as follows. Section 2 contains turnpike results for discretetime Lagrange problems. Section 3 contains preliminaries which we need for the study of the structure of solutions in the regions close to the endpoints. The results of [39], which describe the structure of solutions of Lagrange problems in the regions close to the endpoints, are discussed in Sect. 4. The turnpike results for discrete-time Bolza problems are proved in Sect. 5. Our results (Theorems 15–18 and Propositions 17 and 18) which describe the structure of solutions of Bolza problems in the regions close to the endpoints, are stated in Sect. 6. Section 7 contains a basic lemma for the proof of Theorem 15 which is proved in Sect. 8. Proposition 17 is proved in Sect. 9. Section 10 contains the proof of Theorem 17. Proposition 18 is proved in Sect. 11.

2 The Turnpike Results for Discrete-Time Lagrange Problems Let .X; / be a compact metric space and  be a nonempty closed subset of X  X . Denote by M ./ the set of all bounded functions u W  ! R1 . For each w 2 M ./ set kwk D supfjw.x; y/j W .x; y/ 2 g:

(1)

For each x; y 2 X , each integer T  1 and each u 2 M ./ set .u; T; x/ D supf

T 1 X i D0

u.xi ; xi C1 / W fxi gTiD0 is an ./  program and x0 D xg; (2)

.u; T; x; y/ D supf

T 1 X i D0

u.xi ; xi C1 / W

fxi gTiD0 is an ./  program and x0 D x; xT D yg; .u; T / D supf

T 1 X i D0

u.xi ; xi C1 / W fxi gTiD0 is an ./  programg:

(Here we use the convention that the supremum of an empty set is 1).

(3) (4)

Discrete Time Optimal Control Problems on Large Intervals

95

For each x; y 2 X , each pair of integers T1 ; T2 satisfying 0  T1 < T2 and each 2 1 sequence fut gtTDT  M ./ set 1 2 1 ; T1 ; T2 ; x/ .fut gtTDT 1

D supf

TX 2 1 t DT1

ut .xt ; xt C1 / W

2 fxt gTt DT is an ./  program and xT1 D xg; 1 2 1 .fut gTt DT ; T1 ; T2 ; x; y/ 1

D supf

TX 2 1

t DT1

ut .xt ; xt C1 / W

2 fxt gTt DT is an ./  program and xT1 D x; xT2 D yg; 1 2 1 .fut gtTDT ; T1 ; T2 / 1

D supf

TX 2 1

t DT1

(5)

(6)

2 ut .xt ; xt C1 / W fxt gTt DT is an ./  programg; 1

2 1 .fu O t gtTDT ; T1 ; T2 ; y/ D supf 1

(7) TX 2 1 t DT1

ut .xt ; xt C1 / W

2 fxt gTt DT is an ./  program and xT2 D yg: 1

(8)

Assume that v 2 M ./ is an upper semicontinuous function. We suppose that there exist xN v 2 X and constants cNv > 0 and rNv > 0 such that the following assumptions hold. (A1) f.x; y/ 2 X  X W .x; xN v /; .y; xN v /  rNv g   and v is continuous at .xN v ; xN v /. (A2) .v; T /  T v.xN v ; xN v / C cNv for all integers T  1: Evidently, for each integer T  1 and each ./-program fxt gTtD0 , T 1 X t D0

v.xt ; xt C1 /  .v; T /  T v.xN v ; xN v / C cNv :

Relation (9) implies the following result. Proposition 1. For each ./-program fxt g1 t D0 either the sequence f

T 1 X t D0

v.xt ; xt C1 /  T v.xN v ; xN v /g1 T D1

P 1 v.xt ; xt C1 /  T v.xN v ; xN v / D 1: is bounded or limT !1 Œ tTD0

(9)

96

A.J. Zaslavski

An ./-program fxt g1 t D0 is called .v; /-good if the sequence f

T 1 X t D0

v.xt ; xt C1 /  T v.xN v ; xN v /g1 T D1

is bounded. We suppose that the following assumption holds: (A3) (the asymptotic turnpike property or, briefly, (ATP)) For any .v; /-good program fxt g1 N v / D 0. t D0 , limt !1 .xt ; x

It follows from assumptions (A1) and (A3) that if .xN v ; xN v / is not an isolated point of X  X , then kvk > 0. Examples of optimal control problems satisfying (A1)–(A3) are given in [28, 29, 37]. Denote by Card.A/ the cardinality of a set A and suppose that the sum over empty set is zero. Evidently, for each pair of integers T1 ; T2 satisfying 0  T1 < T2 , each sequence 2 1 fwt gtTDT  M ./ and each x; y 2 X satisfying .x; xN v /; .y; xN v /  rNv the value 1 2 1 .fwt gtTDT ; T1 ; T2 ; x; y/ is finite. 1 Let T be a natural number. We denote by YT the set of all points x 2 X such that there exists an ./-program fxt gTtD0 satisfying x0 D xN v and xT D x and denote by YNT the set of all points x 2 X such that there exists an ./-program fxt gTtD0 such that x0 D x and xT D xN v . The following two theorems obtained in [32] establish the turnpike property for approximate solutions of the optimal control problems of the types (P1) and (P2) with objective functions ut , t D 0; : : : ; T  1 which belong to a small neighborhood of v.

Theorem 1. Let  2 .0; rNv /, L0 be a natural number and M0 > 0. Then there exist an integer L  1 and ı 2 .0; / such that for each integer T > 2L, each 1 fut gtTD0  M ./ satisfying kut  vk  ı; t D 0 : : : T  1; and each ./-program fxt gTtD0 which satisfies x0 2 YNL0 ; xT 2 YL0 ; T 1 X t D0

1 ut .xt ; xt C1 /  .fut gtTD0 ; 0; T; x0 ; xT /  M0

and  CL1 X t D

CL1 ut .xt ; xt C1 /  .fut gtD ; ;  C L; x ; x CL /  ı

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97

for each integer  2 Œ0; T  L there exist integers 1 2 Œ0; L, 2 2 ŒT  L; T  such that .xt ; xN v /  ; t D 1 ; : : : ; 2 : Moreover, if .x0 ; xN v /  ı, then 1 D 0 and if .xT ; xN v /  ı, then 2 D T . Theorem 2. Let a positive number  < rNv , L0  1 be an integer and M0 > 0. Then there exist an integer L  1 and a number ı 2 .0; / such that for each integer 1 T > 2L, each fut gtTD0  M ./ satisfying kut  vk  ı; t D 0 : : : ; T  1 and each ./-program fxt gTtD0 which satisfies x0 2 YNL0 ;

T 1 X t D0

1 ut .xt ; xt C1 /  .fut gtTD0 ; 0; T; x0 /  M0

and  CL1 X t D

CL1 ; ;  C L; x ; x CL /  ı ut .xt ; xt C1 /  .fut gtD

for each integer  2 Œ0; T  L there exist integers 1 2 Œ0; L, 2 2 ŒT  L; T  such that .xt ; xN v /  ; t D 1 ; : : : ; 2 : Moreover if .x0 ; xN v /  ı, then 1 D 0. The next theorem obtained in [36] establishes the turnpike property for approximate solutions of the optimal control problems of the type (P3). Theorem 3. Let  2 .0; rNv / and M > 0. Then there exist a positive number ı < minf1; M g and a natural number L such that the following assertions hold. 1 1. Assume that an integer T  L, fut gtTD0  M ./ and an ./-program fxt gTtD0 satisfy

T 1 X t D0

kut  vk  ı; t D 0; : : : ; T  1;

(10)

1 ut .xt ; xt C1 /  .fut gtTD0 ; 0; T /  M:

(11)

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Then the inequality Card.ft 2 f0; : : : ; T g W .xt ; xN v / > g/ < L holds. 1  M ./ and an ./-program fxt gTtD0 2. Assume that an integer T  2L, fut gtTD0 satisfy (10), (11) and T 1 X t D0

1 ; 0; T; x0 ; xT /  ı: ut .xt ; xt C1 /  .fut gtTD0

(12)

Then there exist integers 1 2 Œ0; L and 2 2 ŒT  L; T  such that .xt ; xN v /  ; t D 1 ; : : : ; 2 : Moreover, if .x0 ; xN v /  ı, then 1 D 0 and if .xT ; xN v /  ı, then 2 D T . 1 3. Assume that fut g1 t D0  M ./ and an ./-program fxt gt D0 satisfy kut  vk  ı for all integers t  0; T 1 X

lim supΠT !1

t D0

1 ut .xt ; xt C1 /  .fut gtTD0 ; 0; T / > M:

Then Card.ft is a nonnegative integer such that .xt ; xN v / > g/ < L: 1  M ./, an ./-program fxt gTtD0 4. Assume that an integer T > 0, fut gtTD0 satisfy (10)–(12) and integers T1 ; T2 satisfy 0  T1 < T2  T . Then TX 2 1

t DT1

2 1 ; T1 ; T2 /  .4L C 2/.2kvk C 2/  M  1: ut .xt ; xt C1 /  .fut gtTDT 1

Assertions 1 and 2 establish turnpike properties for approximate solutions of the problem (P3) while Assertion 3 establish the turnpike property for approximate solutions of the corresponding infinite horizon problem. Moreover, they also show the stability of the turnpike phenomenon under small perturbations of the objective function v. In the sequel we use a notion of an overtaking optimal program [27, 37]. An ./-program fxt g1 t D0 is called .v; /-overtaking optimal if for each ./program fyt g1 satisfying y0 D x0 the inequality t D0

Discrete Time Optimal Control Problems on Large Intervals

lim sup

T 1 X

T !1 t D0

99

Œv.yt ; yt C1 /  v.xt ; xt C1 /  0

holds. The following result obtained in [28] establishes the existence of an overtaking optimal program. Theorem 4. Assume that x 2 X and that there exists a .v; /-good program fxt g1 t D0 such that x0 D x. Then there exists an .v; /-overtaking optimal program  fxt g1 t D0 such that x0 D x. The next result obtained in [28] provides necessary and sufficient conditions for overtaking optimality. Theorem 5. Assume that fxt g1 t D0 is an ./-program and that there exists a .v; /good program fyt g1 such that y0 D x0 . Then the program fxt g1 t D0 t D0 is .v; /overtaking optimal if and only if the following conditions hold: (i) limt !1 .xt ; xN v / D 0; (ii) For each natural number T and each ./-program fyt gTtD0 satisfying y0 D x0 , P 1 P 1 yT D xT the inequality tTD0 v.xt ; xt C1 / holds. v.yt ; yt C1 /  tTD0

The following two theorems obtained in [39] establish the uniform convergence of overtaking optimal programs to xN v .

Theorem 6. Let L0 be a natural number and  be a positive number. Then there exists an integer T0  1 such that for each .v; /-overtaking optimal program N fxt g1 N v /   holds for all integers t  t D0 satisfying x0 2 YL0 the inequality .xt ; x T0 . Theorem 7. Let  > 0. Then there is ı > 0 such that for each .v; /-overtaking N v /  ı the inequality .xt ; xN v /   holds optimal program fxt g1 t D0 satisfying .x0 ; x for all nonnegative integers t. The following two lemmas play an important role in the proofs of the results stated above. Their proofs can be found in [37]. Lemma 1 (Lemma 2.16 of [37]). Let ; M0 be positive numbers. Then there exists an integer T0  1 such that for each natural number T  T0 , each ./-program fxt gTtD0 which satisfies T 1 X t D0

v.xt ; xt C1 /  T v.xN v ; xN v /  M0

and each integer s 2 Œ0; T  T0  the following inequality holds: minf.xi ; xN v / W i D s C 1; : : : ; s C T0 g  :

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Lemma 2 (Lemma 2.15 of [37]). Let  be positive number. Then there exists a number ı 2 .0; rNv / such that for each natural number T and each ./-program fxt gTtD0 which satisfies .x0 ; xN v /; .xT ; xN v /  ı; T 1 X t D0

v.xt ; xt C1 /  .v; x0 ; xT ; T /  ı

the inequality .xt ; xN v /   holds for all integers t D 0; : : : ; T . Lemma 1 shows that if a good program is defined on a sufficiently large interval, then at some point it becomes close to the turnpike. Lemma 2 shows that if an approximate optimal program is close to the turnpike at the endpoints, then it is close to the turnpike at every point. It is clear that .M ./; k
k/ is a Banach space. Denote by M0 ./ the set of all upper semicontinuous functions v 2 M ./ for which there exist xN v 2 X , cNv > 0 and rNv 2 .0; 1/ such that f.x; y/ 2 X  X W .x; xN v /; .y; xN v /  rNv g  ;

(13)

v is continuous at .xN v ; xN v /;

(14)

.v; T /  T v.xN v ; xN v / C cNv for all integers T  1:

(15)

In other words, M0 ./ is the set of all upper semicontinuous functions v 2 M ./ which satisfy assumptions (A1) and (A2) with some xN v 2 X , rNv 2 .0; 1/, cNv > 0. Denote by Mc;0 ./ the set of all continuous functions v 2 M0 ./. Denote by MNc;0 ./ and MN0 ./ the closure of subspaces Mc;0 ./ and M0 ./ in M ./, respectively. We equip the sets MNc;0 ./ and MN0 ./ with the metric d induced by the norm k
k: d.u1 ; u2 / D ku1  u2 k, u1 ; u2 2 MN0 ./. For each u 2 MN0 ./ and each r > 0 set Bd .u; r/ D fw 2 MN0 ./ W ku  wk < rg: We associate with any v 2 M0 ./ the triplet .xN v ; cNv ; rNv / satisfying (13)–(15). Denote by M ./ the set of all v 2 M0 ./ such that for any .v; /-good program fxi g1 i D0 , lim .xi ; xN v / D 0:

i !1

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101

Set Mc ./ D M ./ \ Mc ./: The following theorem was established in [36]. Theorem 8. M ./ contains a set which is a countable intersection of open everywhere dense subsets of MN0 ./ and Mc ./ contains a set which is a countable intersection of open everywhere dense subsets of MNc;0 ./.

3 Preliminaries We use the notation, definitions and assumptions introduced in Sect. 2. Let v 2 M ./ be an upper semicontinuous function. Suppose that xN v 2 X , rNv 2 .0; 1/, cNv > 0 and that assumptions (A1)–(A3) hold. For each M > 0 denote by XM the set of all x 2 X for which there exists a ./-program fxt g1 t D0 such that x0 D x and that for all integers T  1 the following inequality holds: T 1 X t D0

v.xt ; xt C1 /  T v.xN v ; xN v /  M:

Clearly, [fXM W M 2 .0; 1/g is the set of all points x 2 X such that there exists a .v; /-good program fxt g1 t D0 satisfying x0 D x. The boundedness of the function v implies the following useful auxiliary result. Proposition 2. Let T  1 be an integer. Then there exists M > 0 such that YNT  XM . Lemma 1 implies the following useful auxiliary result. Proposition 3. Let M > 0. Then there exists a natural number T such that the inclusion XM  YNT holds.

We define a function  v .x/, x 2 X which plays an important role in our study. For all x 2 X n [fXM W M 2 .0; 1/g set  v .x/ D 1: Let x 2 [fXM W M 2 .0; 1/g:

(16)

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Put  v .x/ D supflim sup

T 1 X

T !1 t D0

.v.xt ; xt C1 /  v.xN v ; xN v // W

ffxt g1 t D0 is an ./  program such that x0 D xg:

(17)

By (A2), (16) and (17),  1 <  v .x/  cNv :

(18)

It follows from (16), (17) and Proposition 1 that  v .x/ D supflim sup

T 1 X

T !1 t D0

.v.xt ; xt C1 /  v.xN v ; xN v // W

fxt g1 t D0 is an .v; /  good program such that x0 D xg:

(19)

Denote by P.v; x/ the set of all .v; /-overtaking optimal programs fxt g1 t D0 such that x0 D x. In view of Theorem 4, the set P.v; x/ is nonempty. Definition (17) implies the following result. Proposition 4. 1. Let fxt g1 t D0 be a .v; /-good program. Then for each integer t  0,  v .xt /  v.xt ; xt C1 /  v.xN v ; xN v / C  v .xt C1 /:

(20)

2. Let T be a natural number and fxt gTtD0 be an ./-program such that  v .xT / > 1. Then inequality (20) is valid for all integers t D 0; : : : ; T  1. The next result follows from the definition of .v; /-overtaking optimal programs. Proposition 5. Let x 2 [fXM W M 2 .0; 1/g and fxt g1 t D0 be a .v; /-overtaking optimal program such that x0 D x. Then  v .x/ D lim sup

T 1 X

T !1 t D0

.v.xt ; xt C1 /  v.xN v ; xN v //:

Corollary 1. Let fxt g1 t D0 be a .v; /-overtaking optimal and .v; /-good program. Then for any integer t  0,  v .xt / D v.xt ; xt C1 /  v.xN v ; xN v / C  v .xt C1 /:

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103

Put sup. v / D supf v .z/ W z 2 [fXM W M 2 .0; 1/gg;

(21)

Xv D fx 2 [fXM W M 2 .0; 1/g W  .x/  sup. /  1g:

(22)

v

v

The next result is proved in [39]. Proposition 6.  v .xN v / D 0. Theorem 7, Proposition 5 and (A1) imply the following result. Proposition 7. The function  v is finite in a neighborhood of xN v and continuous at xN v . The following results are proved in [39]. Proposition 8. Assume that x0 2 [fXM W M 2 .0; 1/g and fxt g1 t D0 2 P.v; x0 /. Then v

 .x0 / D lim

T !1

T 1 X t D0

.v.xt ; xt C1 /  v.xN v ; xN v //:

Proposition 9. There exists a natural number Lv such that Xv  YNLv .

Proposition 10. The function  v W X ! R1 [ f1g is upper semicontinuous. Set D.v/ D fx 2 X W  v .x/ D sup. v /g:

(23)

Proposition 10 and (18) imply that the set D.v/ is nonempty and closed subset of X . The following proposition is also proved in [39]. Proposition 11. Let fxt g1 t D0 be a .v; /-good program such that for all integers t  0, v.xt ; xt C1 /  v.xN v ; xN v / D  v .xt /   v .xt C1 /: Then fxt g1 t D0 is a .v; /-overtaking optimal program. The next result easily follows from Proposition 9, Theorem 6, (22) and (23). Proposition 12. For each  > 0 there exists an integer T  1 such that for each z 2 D.v/ and each ./-program fxt g1 Nv /   t D0 2 P.v; z/ the inequality .xt ; x holds for all integers t  T . In order to study the structure of solutions of the problems (P2) and (P3) we introduce the following notation and definitions.

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Set N D f.x; y/ 2 X  X W .y; x/ 2 g: 

(24)

N is a nonempty closed subset of X  X and It is clear that  N f.x; y/ 2 X  X W .x; xN v /; .y; xN v /  rNv g  :

(25)

N is the set of all bounded functions u W  N ! R1 with Then M ./ N kuk D supfju.z/j W z 2 g: N by For each u 2 M ./ define uN 2 M ./ N uN .x; y/ D u.y; x/; .x; y/ 2 :

(26)

Clearly, u ! uN ; u 2 M ./ is a linear invertible isometry operator. 2 be an ./-program. Define Let 0  T1 < T2 be integers and let fxt gTt DT 1 T2 fxN t gt DT1  X by xN t D xT2 t CT1 ; t D T1 ; : : : ; T2 :

(27)

2 N Clearly, fxN t gTt DT is an ./-program. 1 T2 1 Assume that fut gt DT1  M ./. It is easy to see that

TX 2 1 t DT1

uN T2 t CT1 1 .xN t ; xN t C1 / D

TX 2 1

uT2 t CT1 1 .xT2 t CT1 1 ; xT2 t CT1 /

D

TX 2 1

ut .xt ; xt C1 /:

t DT1

t DT1

The next result easily follows from (28). Proposition 13. Let T  1 be an integer, M  0, 1 fut gtTD0  M ./ .i /

and fxt gTtD0 , i D 1; 2 be ./-programs. Then T 1 X t D0

.1/

.1/

ut .xt ; xt C1 / 

T 1 X t D0

.2/

.2/

ut .xt ; xt C1 /  M

(28)

Discrete Time Optimal Control Problems on Large Intervals

105

if and only if T 1 X t D0

.1/

.1/

uN T t 1 .xN t ; xN t C1 / 

T 1 X t D0

.2/

.2/

uN T t 1 .xN t ; xN t C1 /  M:

Proposition 13 implies the following result. 1  M ./ and fxt gTtD0 Proposition 14. Let T  1 be an integer, M  0, fut gtTD0 T N be an ./-program. Then fxN t gt D0 is an ./-program and the following assertions hold:P 1 1 if tTD0 ut .xt ; xt C1 /  .fut gtTD0 ; 0; T /  M , then T 1 X t D0

if

PT 1 t D0

1 ut .xt ; xt C1 /  .fut gtTD0 ; 0; T; x0 ; xT /  M , then T 1 X t D0

if

PT 1 t D0

1 ; 0; T; xN 0 ; xN T /  M I uN T t 1 .xN t ; xN t C1 /  .fNuT t 1 gtTD0

1 ut .xt ; xt C1 /  O .fut gtTD0 ; 0; T; xT /  M , then T 1 X t D0

if

PT 1 t D0

1 uN T t 1 .xN t ; xN t C1 /  .fNuT t 1 gtTD0 ; 0; T /  M I

1 uN T t 1 .xN t ; xN t C1 /  .fNuT t 1 gtTD0 ; 0; T; xN 0 /  M I

1 ut .xt ; xt C1 /  .fut gtTD0 ; 0; T; x0 /  M , then T 1 X t D0

1 uN T t 1 .xN t ; xN t C1 /  .fN O uT t 1 gtTD0 ; 0; T; xN T /  M:

The next result was obtained in [39]. Proposition 15. Let v 2 M ./ be an upper semicontinuous function. Suppose that xN v 2 X , rNv > 0, cNv > 0 and that assumptions (A1)–(A3) hold. Then the function vN is upper semicontinuous, N f.x; y/ 2 X  X W .x; xN v /; .y; xN v /  rNv g  ;

(29)

the function vN is continuous at .xN v ; xN v /, .v; N T /  T v. N xN v ; xN v / C cNv for all integers T  1

(30)

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N and for all .v; N /-good programs fxt g1 t D0 , lim .xt ; xN v / D 0:

t !1

In view of Proposition 15, if v 2 M ./ is upper semicontinuous and satisfies (A1)–(A3), then vN is also upper semicontinuous and satisfies (A1)–(A3). Therefore N all the results presented above for the pair .v; / are also true for the pair .v; N /.

4 Structure of Solutions of Lagrange Problems in the Regions Close to the Endpoints We use the notation, definitions and assumptions introduced in Sects. 2 and 3. The results of this session were obtained in [39]. The following result describes the structure of approximate solutions of the problems of the type (P2) in the regions close to the right endpoints. Theorem 9. Suppose that v 2 M ./ is an upper semicontinuous function, xN v 2 X , rNv > 0, cNv > 0 and that assumptions (A1)–(A3) hold. Let L0  1, 0  1 be integers and  > 0. Then there exist ı > 0 and an integer T0  0 such that for each 1 integer T  T0 , each fut gtTD0  M ./ satisfying kut  vk  ı; t D 0 : : : ; T  1 and each ./-program fxt gTtD0 which satisfies x0 2 YNL0 ;

T 1 X t D0

1 ut .xt ; xt C1 /  .fut gtTD0 ; 0; T; x0 /  ı

N there exists an ./-program fxt g1 N z/ W z 2 D.v/g N t D0 2 [fP.v; such that .xT t ; xt /  ; t D 0; : : : ; 0 : N Recall that [fP.v; N z/ W z 2 D.v/g N is the set of all .v; N /-overtaking optimal  1  programs fxt gt D0 such that x0 is the point of maximum of the function  vN . The following result describes the structure of approximate solutions of the problems of the type (P3) in the regions close to the endpoints.

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107

Theorem 10. Suppose that v 2 M ./ is an upper semicontinuous function, xN v 2 X , rNv > 0, cNv > 0 and that assumptions (A1)–(A3) hold. Let 0  1 be an integer and  > 0. Then there exist ı > 0 and an integer T0  0 such that for each integer 1 T  T0 , each fut gtTD0  M ./ satisfying kut  vk  ı; t D 0 : : : ; T  1 and each ./-program fxt gTtD0 which satisfies T 1 X t D0

1 ut .xt ; xt C1 /  .fut gtTD0 ; 0; T /  ı

there exist an ./-program fyt g1 t D0 2 [fP.v; z/ W z 2 D.v/g N and an ./-program fxt g1 N z/ W z 2 D.v/g N t D0 2 [fP.v; such that for all integers t D 0; : : : ; 0 ; .xT t ; xt /  ; .xt ; yt /  : Proposition 16. Suppose that v 2 M ./ is an upper semicontinuous function, xN v 2 X , rNv > 0, cNv > 0 and that assumptions (A1)–(A3) hold. Let 0  1 be an integer and  > 0. Then there exist ı > 0 and an integer T0  0 such that for each u 2 Bd .v; r/ \ M ./ the following properties hold: for each fxt g1 t D0 2 [fP.u; z/ W z 2 D.u/g there exists fyt g1 t D0 2 [fP.v; z/ W z 2 D.v/g such that .xt ; yt /   for all integers t D 0; : : : ; 0 ; u; z/ W z 2 D.Nu/g there exists for each fxt g1 t D0 2 [fP.N fyt g1 N z/ W z 2 D.v/g N t D0 2 [fP.v; such that .xt ; yt /   for all integers t D 0; : : : ; 0 . We have already mentioned that the mapping v ! v, N v 2 M ./ is a linear isometry which has the inverse. It is not difficult to see that N for all v 2 M0 ./; vN 2 M0 ./

N for all v 2 Mc;0 ./; vN 2 Mc;0 ./

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N for all v 2 M ./; vN 2 M ./

N for all v 2 Mc ./; vN 2 Mc ./

N if and only if v 2 MN0 ./; vN 2 MN0 ./

N if and only if v 2 MNc;0 ./: vN 2 MNc;0 ./ The next theorem shows that for most objective functions v (in the sense of the Baire category) the sets [fP.v; z/ W z 2 D.v/g and [fP.v; N z/ W z 2 D.v/g N are singletons. In this case approximate solutions of the problems of the types (P2) and (P3) in the regions close to the endpoints have a simple structure. Theorem 11. Let M be either MN0 ./ or MNc;0 ./. Then there exists a set F  M \ M ./ which is a countable intersection of open everywhere dense subsets of M such that for each v 2 F there exists a unique pair of points z; zN 2 X such that  v .z/ D sup. v /;  vN .Nz/ D sup. vN / and there exist a unique .v; /-overtaking optimal program fzt g1 t D0 satisfying z0 D N z and a unique .v; N /-overtaking optimal program fb zt g1 satisfyingb z0 D zN. t D0

5 The Turnpike Results for Discrete-Time Bolza Problems We use the notation, definitions and assumptions introduced in Sects. 2 and 3. For each nonempty set Y and each function h W Y ! R1 [ f1g put sup.h/ D supfh.y/ W y 2 Y g: Denote by M .X / the set of all bounded functions h W X ! R1 . For each h 2 M .X / set khk D supfjh.x/j W x 2 X g: Clearly, .M .X /; k
k/ is a Banach space. For each h1 ; h2 2 M .X / set dX .h1 ; h2 / D kh1  h2 k: For each x 2 X , each pair of integers T1 ; T2 satisfying 0  T1 < T2 , each sequence 2 1 fut gTt DT  M ./ and each h 2 M .X / we consider the problem 1 TX 2 1

t DT1

1 ut .xt ; xt C1 / C h.xT2 / ! max; f.xt ; xt C1 /gtTD0  ; xT1 D x

(P4)

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109

and set 2 1 .h; fut gtTDT ; T1 ; T2 ; x/ D supf 1

TX 2 1 t DT1

ut .xt ; xt C1 / C h.xT2 / W

2 fxt gTt DT is an ./  program and xT1 D xg: 1

(31)

For each x 2 X , each pair of integers T1 ; T2 satisfying 0  T1 < T2 , each u 2 M ./ and each h 2 M .X / set 2 1 .h; u; T1 ; T2 ; x/ D .h; fut gtTDT ; T1 ; T2 ; x/ where ut D u; t D T1 ; : : : ; T2  1: 1 (32)

In this section we study turnpike properties of approximate solutions of problems of the type (P4) and establish the following three results. Theorem 12. Let  2 .0; rNv / and M > 0. Then there exist a positive number ı < minf1; M g and a natural number L such that for each integer T  L, each 1 fut gtTD0  M ./, each h 2 M .X / and each ./-program fxt gTtD0 which satisfy T 1 X t D0

khk  M; kut  vk  ı; t D 0; : : : ; T  1;

(33)

1 ut .xt ; xt C1 / C h.xT /  .h; fut gtTD0 ; 0; T /  M

(34)

Card.ft 2 f0; : : : ; T g W .xt ; xN v / > g/ < L

(35)

the inequality

holds. Proof. By Assertion 1 of Theorem 3, there exist a positive number ı < minf1; M g and a natural number L such that the following property holds: 1  M ./ and each ./-program fxt gTtD0 (i) For each integer T  L, each fut gtTD0 which satisfy

kut  vk  ı; t D 0; : : : ; T  1; T 1 X t D0

inequality (35) holds.

1 ut .xt ; xt C1 /  .fut gtTD0 ; 0; T /  8M

(36)

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1 Assume that an integer T  L, fut gtTD0  M ./, h 2 M .X /, khk  M , T fxt gt D0 is an ./-program and (33) and (34) hold. In view of (33) and (34), inequality (36) holds. By (33), (36) and property (i), the inequality (35) holds. Theorem 12 is proved.

Theorem 13. Let a positive number  < rNv , L0  1 be an integer and M0 > 0. Then there exist an integer L  1 and a number ı 2 .0; / such that for each integer 1 T > 2L, each fut gtTD0  M ./ satisfying kut  vk  ı; t D 0 : : : ; T  1;

(37)

each h 2 M .X / satisfying khk  M0

(38)

and each ./-program fxt gTtD0 which satisfies x0 2 YNL0 ; h.xT / C

T 1 X t D0

1 ut .xt ; xt C1 /  .h; fut gtTD0 ; 0; T; x0 /  M0

(39) (40)

and  CL1 X t D

CL1 ut .xt ; xt C1 /  .fut gtD ; ;  C L; x ; x CL /  ı

(41)

for each integer  2 Œ0; T  L there exist integers 1 2 Œ0; L, 2 2 ŒT  L; T  such that .xt ; xN v /  ; t D 1 ; : : : ; 2 :

(42)

Moreover if .x0 ; xN v /  ı, then 1 D 0. Proof. By Theorem 2, there exist an integer L  1 and a number ı 2 .0; / such that the following property holds: 1 (ii) For each integer T > 2L, each fut gtTD0  M ./ satisfying (37) and each T ./-program fxt gt D0 which satisfies (39), the inequality T 1 X t D0

1 ; 0; T; x0 /  4M0 ut .xt ; xt C1 /  .fut gtTD0

(43)

and (41) for each integer  2 Œ0; T  L, there exist integers 1 2 Œ0; L, 2 2 ŒT  L; T  such that (42) holds; moreover if .x0 ; xN v /  ı, then 1 D 0.

Discrete Time Optimal Control Problems on Large Intervals

111

1 Assume that an integer T > 2L, fut gtTD0  M ./ satisfies (37), h 2 M .X / satisfies (38) and that an ./-program fxt gTtD0 satisfies (39)–(41) for each integer  2 Œ0; T  L. In view of (38) and (40), relation (43) holds. By (37), (39), (41), (43) and property (ii), there exist integers 1 2 Œ0; L and 2 2 ŒT  L; T  such that (42) holds and if .x0 ; xN v /  ı, then 1 D 0. Theorem 13 is proved.

Theorem 13 implies the following turnpike result. Theorem 14. Let a positive number  < rNv , L0  1 be an integer and M0 > 0. Then there exist an integer L  1 and a number ı 2 .0; / such that for each integer 1 T > 2L, each fut gtTD0  M ./ satisfying kut  vk  ı; t D 0 : : : ; T  1; each h 2 M .X / satisfying khk  M0 and each ./-program fxt gTtD0 which satisfies x0 2 YNL0 ; h.xT / C

T 1 X t D0

1 ut .xt ; xt C1 /  .h; fut gtTD0 ; 0; T; x0 /  ı

there exist integers 1 2 Œ0; L, 2 2 ŒT  L; T  such that .xt ; xN v /  ; t D 1 ; : : : ; 2 : Moreover if .x0 ; xN v /  ı, then 1 D 0.

6 Structure of Solutions of Bolza Problems in the Regions Close to the Endpoints We use the notation, definitions and assumptions introduced in Sects. 2, 3 and 5. In Sect. 8 we prove the following result which describes the structure of approximate solutions of the problems of the type (P4) in the regions close to the right endpoints. Theorem 15. Suppose that g 2 M .X / and v 2 M ./ are upper semicontinuous functions, xN v 2 X , rNv > 0, cNv > 0 and that assumptions (A1)–(A3) hold. Let L0  1, 0  1 be integers and  > 0, M0 > 1. Then there exist ı > 0 and an integer T0  0 such that for each integer T  T0 , each h 2 M .X / satisfying kh  gk  ı;

112

A.J. Zaslavski

1 each fut gtTD0  M ./ satisfying

kut  vk  ı; t D 0 : : : ; T  1 and each ./-program fxt gTtD0 which satisfies x0 2 YNL0 ; h.xT / C  CT 0 1 X t D

T 1 X t D0

1 ut .xt ; xt C1 /  .h; fut gtTD0 ; 0; T; x0 /  M0 ;

CT0 1 ; ;  C T0 ; x ; x CT0 /  ı ut .xt ; xt C1 /  .fut gt D

for each  2 f0; : : : ; T  T0 g, h.xT / C

T 1 X

t DT T0

1 ut .xt ; xt C1 /  .h; fut gtTDT T0 ; T  T0 ; T; xT T0 /  ı

N there exists an .v; N /-overtaking optimal program fxt g1 t D0 such that . vN C g/.x0 / D sup. vN C g/;

.xT t ; xt /  ; t D 0; : : : ; 0 : Let g 2 M .X / and v 2 M ./ be as in the statement of Theorem 15 and satisfy all the assumptions posed there. Then  v C g W X ! R1 [ f1g is an upper semicontinuous, bounded from above function such that . v C g/.xN v / D g.xN v / is finite. Therefore there exists x 2 X such that . v C g/.x/ D sup. v C g/: Denote by Mu .X / the set of all upper semicontinuous functions belonging to the space M .X / and denote by Mc .X / the set of all continuous functions belonging to the space M .X /. Clearly, Mu .X / and Mc .X / are closed subsets of M .X /. We consider the complete metric spaces Mu .X / and Mc .X / equipped with the metric dX . In order to state our next result we need the following notion of porosity [35]. Let .Y; d / be a complete metric space. We denote by BY .y; r/ the closed ball of center y 2 Y and radius r > 0. A subset E  Y is called porous (with respect to d ) if there exist ˛ 2 .0; 1 and r0 > 0 such that for each r 2 .0; r0  and each y 2 Y there exists z 2 Y for which BY .z; ˛r/  BY .y; r/ n E:

Discrete Time Optimal Control Problems on Large Intervals

113

A subset of the space Y is called -porous (with respect to d ) if it is a countable union of porous (with respect to d ) subsets of Y . Since porous sets are nowhere dense, all -porous sets are of the first category. If Y is a finite dimensional Euclidean space, then -porous sets are of Lebesgue measure 0. In fact, the class of -porous sets in such a space is much smaller than the class of sets which have measure 0 and are of the first category. To point out the difference between porous and nowhere dense sets note that if E  Y is nowhere dense, y 2 Y and r > 0, then there is a point z 2 Y and a number s > 0 such that BY .z; s/  BY .y; r/ n E. If, however, E is also porous, then for small enough r we can choose s D ˛r, where ˛ 2 .0; 1/ is a constant which depends only on E. The discussion of the porosity notion and the corresponding references can be found in [35]. Theorem 5.9 of [35] and Theorem 15 imply the following result. Theorem 16. Suppose that v 2 M ./ is an upper semicontinuous function, xN v 2 X , rNv > 0, cNv > 0 and that assumptions (A1)–(A3) hold. Then there exists a set F  Mc ./ such that the set Mc ./ n F is -porous in Mc ./ and that for each g 2 F the following assertions hold. 1. There exists a unique point xg 2 X such that fx 2 X W . vN C g/.x/ D sup. vN C g/g D fxg g: 2. Let L0  1, 0  1 be integers and  > 0, M0 > 1. Then there exist ı > 0 and an integer T0  0 such that for each integer T  T0 , each h 2 M .X / satisfying kh  gk  ı; 1  M ./ satisfying each fut gtTD0

kut  vk  ı; t D 0 : : : ; T  1 and each ./-program fxt gTtD0 which satisfies x0 2 YNL0 ; h.xT / C  CT 0 1 X t D

T 1 X t D0

1 ; 0; T; x0 /  M0 ; ut .xt ; xt C1 /  .h; fut gtTD0

CT0 1 ; ;  C T0 ; x ; x CT0 /  ı ut .xt ; xt C1 /  .fut gtD

for each  2 f0; : : : ; T  T0 g, h.xT / C

T 1 X

t DT T0

1 ut .xt ; xt C1 /  .h; fut gtTDT T0 ; T  T0 ; T; xT T0 /  ı

114

A.J. Zaslavski

N there exists an .v; N /-overtaking optimal program fxt g1 t D0 such that x0 D xg ;

.xT t ; xt /  ; t D 0; : : : ; 0 : The next theorem which is proved in Sect. 10 shows that given a function g 2 Mu .X /, for most objective functions v (in the sense of the Baire category) the N exists a unique pair of a .v; /-overtaking optimal program fzt g1 N /t D0 and a .v; 1 overtaking optimal program fb zt gt D0 such that . v C g/.z0 / D sup. v C g/; . vN C g/.b z0 / D sup. vN C g/: In this case approximate solutions of the problems of the types (P4) in the regions close to the right endpoints have a simple structure. Theorem 17. Let M be either MN0 ./ or MNc;0 ./ and let g 2 Mu .X /. Then there exists a set F  M \ M ./ which is a countable intersection of open everywhere dense subsets of M such that for each v 2 F there exists a unique pair of points z; zN 2 X such that .g C  v /.z/ D sup.g C  v /; .g C  vN /.Nz/ D sup.g C  vN / and there exist a unique .v; /-overtaking optimal program fzt g1 t D0 satisfying z0 D N z and a unique .v; N /-overtaking optimal program fb zt g1 satisfyingb z0 D zN. t D0 Let v 2 M ./ and g 2 Mu .X /. Set D.g; v/ D fz 2 X W .g C  v /.z/ D sup.g C  v /g Q and denote by P.g; v/ the set of all .v; /-overtaking optimal programs fzt g1 t D0 satisfying z0 2 D.g; v/. In Sect. 9 we prove the following result. Proposition 17. Suppose that g 2 Mu .X / and v 2 M ./ are upper semicontinuous functions, xN v 2 X , rNv > 0, cNv > 0 and that assumptions (A1)–(A3) hold. Let 0  1 be an integer and  > 0. Then there exist ı > 0 and an integer T0  0 such that for each u 2 Bd .v; r/ \ M ./ and each h 2 Mu .X / satisfying kh  gk  ı the following properties hold: 1 Q Q for each fxt g1 t D0 2 P.h; u/ there exists fyt gt D0 2 P.g; v/ such that .xt ; yt /   for all integers t D 0; : : : ; 0 ;

Discrete Time Optimal Control Problems on Large Intervals

115

Q Q for each fxt g1 N / there exists fyt g1 N such that t D0 2 P.h; u t D0 2 P.g; v/ .xt ; yt /   for all integers t D 0; : : : ; 0 . The next result is an extension of Theorem 17. Theorem 18. Let M be either MN0 ./ or MNc;0 ./ and A be either Mu .X / or Mc .X /. Then there exists a set F  .M \ M .//  A which is a countable intersection of open everywhere dense subsets of M  A such that for each Q .v; g/ 2 F there exists a unique pair of programs fxt g1 t D0 2 P.g; v/ and 1 Q N v/. fxN t gt D0 2 P.g; Since the mapping v ! v, N v 2 M ./ is a linear isometry which has the inverse, Theorem 18 follows from the next proposition which is proved in Sect. 11. Proposition 18. Let M be either MN0 ./ or MNc;0 ./ and A be either Mu .X / or Mc .X /. Then there exists a set F  .M \ M .//  A which is a countable intersection of open everywhere dense subsets of M  A such that for each .v; g/ 2 Q F there exists a unique program fxt g1 t D0 2 P.g; v/.

7 A Basic Lemma for Theorem 15 Lemma 3. Suppose that g 2 M .X / and v 2 M ./ are upper semicontinuous functions, xN v 2 X , rNv > 0, cNv > 0 and that assumptions (A1)–(A3) hold. Let T0  1 be an integer and  2 .0; 1/. Then there exists ı 2 .0; / such that for each ./0 program fxt gTt D0 which satisfies . v C g/.x0 /  sup. v C g/  ı; TX 0 1 t D0

.v.xt ; xt C1 /  v.xN v ; xN v //   v .x0 / C  v .xT0 /  ı

(44) (45)

there exists an .v; /-overtaking optimal program fzt g1 t D0 such that . v C g/.z0 / D sup. v C g/;

(46)

.zt ; xt /  ; t D 0; : : : ; T0 :

(47)

Proof. Assume that the lemma does not hold. Then there exist a sequence of real .k/ T0 numbers fık g1 kD1  .0; 1 and a sequence of ./-programs fxt gt D0 , k D 1; 2; : : : such that lim ık D 0

k!1

(48)

116

A.J. Zaslavski

and that for each natural number k and each .v; /-overtaking optimal program fzt g1 t D0 satisfying (46), .k/

TX 0 1 t D0

. v C g/.x0 /  sup. v C g/  ık ;

(49)

.k/

(50)

maxf.zt ; xt / W t D 0; : : : ; T0 g > :

(51)

.k/

.k/

.k/

.v.xt ; xt C1 /  v.xN v ; xN v //   v .x0 / C  v .xT0 /  ık ; .k/

.k/

.k/

By (49) and (50), for each natural numbers k, the values  v .x0 / and  v .xT0 / are finite. Extracting a subsequence and re-indexing, if necessary, we may assume without loss of generality that for each integer t 2 Œ0; T0  there exists .k/

xt D lim xt :

(52)

k!1

By (52) and upper semicontinuity of g and  v , .k/

.k/

 v .x0 /  lim sup  v .x0 /; g.x0 /  lim sup g.x0 /:

(53)

k!1

k!1

It follows from Proposition 10, (49) and (53) that .k/

sup. v C g/  . v C g/.x0 /  lim sup. v C g/.x0 /  sup. v C g/;

(54)

k!1

.k/

sup. v C g/ D . v C g/.x0 / D lim . v C g/.x0 /:

(55)

k!1

Relations (53) and (55) imply that .k/

.k/

 v .x0 / D lim  v .x0 /; g.x0 / D lim g.x0 /:

(56)

k!1

k!1

It follows from upper semicontinuity of v and  v (see Proposition 10), (48), (50), (52) and (56) that TX 0 1 t D0

 lim sup. k!1

.v.xt ; xt C1 /  v.xN v ; xN v //   v .x0 / C  v .xT0 /

TX 0 1 t D0

.k/

.k/

.k/

.k/

.v.xt ; xt C1 /  v.xN v ; xN v //   v .x0 / C  v .xT0 //  lim sup.ık / D 0: k!1

(57)

Discrete Time Optimal Control Problems on Large Intervals

117

Relations (55) and (57) imply that the values  v .x0 / and  v .xT0 / are finite. Combined with Proposition 4 and (57) this implies that for all integers t D 0; : : : ; T0  1, v.xt ; xt C1 /  v.xN v ; xN v / D  v .xt /   v .xt C1 /:

(58)

Since  v .xT0 / is finite it follows from Theorem 4 that there exists a .v; /overtaking optimal and .v; /-good program fxQ t g1 t D0 satisfying xQ 0 D xT0 :

(59)

xt D xQ t T0 :

(60)

For all integers t > T0 set

Q t g1 Evidently, fxt g1 t D0 is an ./-program. Since fx t D0 is .v; /-overtaking optimal and .v; /-good program it follows from Corollary 1 and (60) that (58) holds for all integers t  0. Since fxQ t g1 t D0 is .v; /-good program relations (58), (59) and Proposition 11 imply that fxt g1 t D0 is .v; /-overtaking optimal program satisfying .k/ (55). By (52), for all sufficiently large natural numbers k, .xt ; xt /  =4, t D 0; : : : ; T0 . This contradicts (46) and (51). The contradiction we have reached proves Lemma 3.

8 Proof of Theorem 15 We may assume that rNv 2 .0; 1/. Recall that f.x; y/ 2 X  X W .x; xN v /; .y; xN v /  rNv g  :

(61)

By Lemma 3 applied to the function vN there exists a real number ı1 2 .0; minf; rNv =2g/ such that the following property holds: (Pi)

0 N which satisfies For each ./-program fyt gt D0

. vN C g/.y0 /  sup. vN C g/  ı1 ; X 0 1 t D0

.v.y N t ; yt C1 /  v. N xN v ; xN v //   vN .y0 / C  vN .y0 /  ı1

(62) (63)

118

A.J. Zaslavski

N there exists an .v; N /-overtaking optimal program fzt g1 t D0 such that . vN C g/.z0 / D sup. vN C g/;

(64)

.zt ; yt /  ; t D 0; : : : ; 0 :

(65)

Propositions 6 and 7 and assumption (A1) imply that there exists a real number ı2 2 .0; ı1 / such that for each point z 2 X satisfying .z; xN v /  2ı2 , j vN .z/j D j vN .z/   vN .xN v /j  ı1 =8

(66)

and for each .x; y/ 2 X  X satisfying .x; xN v /  2ı2 , .y; xN v /  2ı2 , jv.x; y/  v. N xN v ; xN v /j  ı1 =8:

(67)

It follows from Theorem 13 that there exist a natural number L and a positive number ı3 such that the following property holds: (Pii)

1  M ./ satisfying For each integer T > 2L, each fut gtTD0

kut  vk  ı3 ; t D 0 : : : ; T  1; each h 2 M .X / satisfying kh  gk  1 and each ./-program fzt gTtD0 which satisfies z0 2 YNL0 ; h.zT / C  CL1 X t D

T 1 X t D0

1 ut .zt ; zt C1 /  .h; fut gtTD0 ; 0; T; z0 /  M0 ;

CL1 ; ;  C L; z ; z CL /  ı3 ut .zt ; zt C1 /  .fut gt D

for each integer  2 Œ0; T  L, we have .zt ; xN v /  ı2 ; t D L; : : : ; T  L: Set D0 D fz 2 X W . vN C g/.z/ D sup. vN C g/g:

(68)

Proposition 6, (18) and (25) imply that the set D0 is nonempty and closed. Let z 2 D0 :

(69)

Discrete Time Optimal Control Problems on Large Intervals

119

By (68) and (69),  vN .z/  sup. vN C g/  kgk:

(70)

N N /-overtaking optimal It follows from (70) and Theorem 4 that there exists a .v; 1 N and .v; N /-good program fzt gt D0 such that z0 D z 2 D0 :

(71)

N By Proposition 8, for each .v; N /-overtaking optimal program ft g1 t D0 such that 0 2 D0 we have  vN .0 / D lim

T !1

T 1 X t D0

.v. N t ; t C1 /  v. N xN v ; xN v //:

(72)

Lemma 1, (70) and (72) imply that there exists a natural number L1 such that D0  YL1 :

(73)

By (73) and Theorem 6 applied to the function vN there exists a natural number 1 N such that for each .v; N /-overtaking optimal program fzt g1 t D0 satisfying z0 2 D0 we have .zt ; xN v /  ı2 for all integers t  1 :

(74)

Choose a number ı > 0 and a natural number T0 such that ı < .16.L C 1 C 0 C 8//1 minfı1 ; ı2 ; ı3 g;

(75)

T0 > 2L C 20 C 21 C 4:

(76)

Assume that an integer T  T0 , h 2 M .X / satisfies kh  gk  ı;

(77)

kut  vk  ı; t D 0 : : : ; T  1

(78)

1 fut gtTD0  M ./ satisfies

and fxt gTtD0 is an ./-program which satisfies x0 2 YNL0 ; h.xT / C

T 1 X t D0

1 ; 0; T; x0 /  M0 ; ut .xt ; xt C1 /  .h; fut gtTD0

(79) (80)

120

A.J. Zaslavski  CT 0 1 X t D

CT0 1 ; ;  C T0 ; x ; x CT0 /  ı ut .xt ; xt C1 /  .fut gtD

(81)

for each integer  2 Œ0; T  T0 , h.xT / C

T 1 X

t DT T0

1 ut .xt ; xt C1 /  .h; fut gtTDT T0 ; T  T0 ; T; xT T0 /  ı:

(82)

It follows from (75) to (79), (81) and the property (Pii) that .xt ; xN v /  ı2 ; t D L; : : : ; T  L:

(83)

By (76), ŒT  L  0  1  4; T  L  0  1   ŒL; T  L  0  1 :

(84)

In view of (83) and (84), .xt ; xN v /  ı2 ; t 2 fT  L  0  1  4; T  L  0  1 g:

(85)

N N /-overtaking optimal program fzt g1 There exists (see (68)–(71)) an .v; t D0 such that z0 2 D0 ; . vN C g/.z0 / D sup. vN C g/:

(86)

Then (74) holds. Define a sequence fxQ t gTtD0 by xQ t D xt ; t D 0; : : : ; T  L  0  1  4; xQ t D zT t ; t D T  L  0  1  3; : : : ; T:

(87)

Relations (74) and (87) imply that .xQ T L0 1 3 ; xN v / D .zLC0 C1 C3 ; xN v /  ı2 :

(88)

It follows from (61), (85), (87) and (88) that fxQ t gTtD0 is an ./-program. By (76)– (78), (82) and (87), ı

1 .h; fut gtTDT T0 ; T

 h.xQ T / C

T 1 X

t DT T0

 T0 ; T; xT T0 /  h.xT / 

ut .xQ t ; xQ t C1 /  h.xT / 

T 1 X

t DT T0

T 1 X

ut .xt ; xt C1 /

t DT T0

ut .xt ; xt C1 /

Discrete Time Optimal Control Problems on Large Intervals

D h.xQ T /  h.xT / C

T 1 X

t DT L0 1 4

 g.xQ T /  g.xT / C 

T 1 X

t DT L0 1 4

121

.ut .xQ t ; xQ t C1 /  ut .xt ; xt C1 //

T 1 X

t DT L0 1 4

v.xQ t ; xQ t C1 /

v.xt ; xt C1 /  2ı.L C 0 C 1 C 6/:

Combined with (85) this implies that g.xT / C  g.xQ T / C

T 1 X

v.xt ; xt C1 /

t DT L0 1 4

T 1 X

t DT L0 1 4

v.xQ t ; xQ t C1 /  2ı.L C 0 C 1 C 8/

D g.z0 / C v.xT L0 1 4 ; zLC0 C1 C3 / C

LCX 0 C1 C2 t D0

v.z N t ; zt C1 /  2ı.L C 0 C 1 C 8/:

(89)

In view of (67), (74) and (85), jv.z N LC0 C1 C3 ; xT L0 1 4 /  v.z N LC0 C1 C3 ; zLC0 C1 C4 /j  ı1 =4: Combined with (89) this implies that g.xT / C  g.z0 / C

LCX 0 C1 C3 t D0

T 1 X

v.xt ; xt C1 /

t DT L0 1 4

v.z N t ; zt C1 /  ı1 =4  2ı.L C 0 C 1 C 8/:

(90)

Set yt D xT t ; t D 0; : : : ; L C 0 C 1 C 4: By (75), (90) and (91), g.y0 / C

LCX 0 C1 C3 t D0

v.y N t ; yt C1 /

(91)

122

A.J. Zaslavski

D g.xT / C D g.xT / C  g.z0 / C

LCX 0 C1 C3

v.xT t 1 ; xT t /

t D0

T 1 X

v.xt ; xt C1 /

t DT L0 1 4

LCX 0 C1 C3 t D0

v.z N t ; zt C1 /  3ı1 =8:

(92)

N N /-overtaking optimality of By (86), (92), Proposition 4 and Corollary 1 and .v; 1 fzt gt D0 ; . vN C g/.y0 /  sup. vN C g/ X 0 1

C

C

t D0

.v.y N t ; yt C1 /  v. N xN v ; xN v //   vN .y0 / C  vN .y0 /  . vN C g/.y0 /  . vN C g/.z0 /

LCX 0 C1 C3

.v.y N t ; yt C1 /  v. N xN v ; xN v //   vN .y0 / C  vN .yLC0 C1 C4 /

t D0

vN

vN

  .y0 /   .z0 / C

LCX 0 C1 C3 t D0

.v.z N t ; zt C1 /  v. N xN v ; xN v //

 vN .y0 / C  vN .yLC0 C1 C4 /  3ı1 =8 D  vN .zLC0 C1 C4 /   vN .yLC0 C1 C4 /  3ı1 =8:

(93)

In view of (66) and (74),  vN .zLC0 C1 C4 /  ı1 =8: By (66), (85) and (91), j vN .yLC0 C1 C4 /j D j vN .xT L0 1 4 /j  ı1 =8: Combined with (93) and (94) this implies that . vN C g/.y0 /  sup. vN C g/ C

X 0 1 t D0

.v.y N t ; yt C1 /  v. N xN v ; xN v //   vN .y0 / C  vN .y0 /  ı1 :

(94)

Discrete Time Optimal Control Problems on Large Intervals

123

Together with (83) and Proposition 4 this implies that . vN C g/.y0 /  sup. vN C g/  ı1 ; X 0 1 t D0

.v.y N t ; yt C1 /  v. N xN v ; xN v //   vN .y0 / C  vN .y0 /  ı1 :

(95) (96)

N It follows from (95), (96) and the property (Pi) that there exists an .v; N /-overtaking 1 optimal program ft gt D0 such that . vN C g/.0 / D sup. vN C g/; .t ; xT t / D .t ; yt /  ; t D 0; : : : ; 0 : Theorem 15 is proved.

t u

9 Proof of Proposition 17 Since the mapping v ! v, N v 2 M ./ is an isometry Proposition 17 follows from Proposition 15 and the following result. Proposition 19. Suppose that g 2 Mu .X / and v 2 M ./ is an upper semicontinuous function, xN v 2 X , rNv > 0, cNv > 0 and that assumptions (A1)–(A3) hold. Let 0  1 be an integer and  > 0. Then there exist ı > 0 and an integer T0  0 such that for each u 2 Bd .v; ı/ \ M ./ and each h 2 Mu .X / satisfying kh  gk  ı the following property holds: Q Q for each fxt g1 N / there is fyt g1 N such that .xt ; yt /   t D0 2 P.h; u t D0 2 P.g; v/ for all integers t D 0; : : : ; 0 . Proof. By (A2), for each u 2 M0 ./ and each integer T  1, .u; T; xN u ; xN u / D T u.xN u ; xN u /;

(97)

u.xN u ; xN u /  u.z; z/ for all z 2 X such that .z; z/ 2 :

(98)

We show that together with (98) and Assertion 1 of Theorem 3 this implies that there exists ı0 2 .0; / such that for each u 2 Bd .v; ı0 / \ M0 ./, .xN u ; xN v /  rNv =4:

(99)

By Assertion 1 of Theorem 3, there exist a positive number ıQ < 1 and a natural number LQ such that the following property holds: Q each fut gT 1  M ./ and each ./-program fxt gT (i) For each integer T  L, t D0 t D0 which satisfy Q t D 0; : : : ; T  1; kut  vk  ı;

124

A.J. Zaslavski T 1 X t D0

1 ; 0; T /  2cNv  4 ut .xt ; xt C1 /  .fut gtTD0

we have Q Card.ft 2 f0; : : : ; T g W .xt ; xN v / > 81 rNv g/ < L: Choose a positive number ı0 such that Q ı0 < ; ı0 < .2L/ Q 1 : ı0 < ı;

(100)

u 2 Bd .v; ı0 / \ M0 ./:

(101)

Assume that

In view of (101), relation (98) holds. Set t D xN u ; t D 0; 1; : : : :

(102)

We show that (99) holds. Assume the contrary. Then .t ; xN v / > rNv =4; t D 0; 1; : : : :

(103)

By (100), (101), (103) and property (i), for each integer k  0, Q .kC1/L1

X

Q t Dk L

Q  2cNv  4: u.t ; t C1 / < .u; 0; L/

(104)

It follows from (100), (101) and (A2) that Q xN v ; xN v / C cNv C ı0 LQ Q  .v; 0; L/ Q C ı0 LQ  Lv. .u; 0; L/

Q xN v ; xN v / C cNv C 2ı0 LQ  Lu. Q xN v ; xN v / C cNv C 1:  Lu. Relations (104) and (105) imply that for each integer k  0, Q .kC1/L1

X

Q t Dk L

Q xN v ; xN v /  cNv  3: u.t ; t C1 /  Lu.

Together with (102) this implies that Q xN u ; xN u /  Lu. Q xN v ; xN v /  cNv  3; Lu.

(105)

Discrete Time Optimal Control Problems on Large Intervals

125

u.xN u ; xN u / < u.xN v ; xN v /: This contradicts (98). The contradiction we have reached proves (99). Thus we have shown that for each function u satisfying (101) relation (99) is valid. By Theorem 15, there exist ı 2 .0; ı0 / and a natural number T0  0 such that the following property holds: (ii) For each integer T  T0 , each h 2 M .X / satisfying kh  gk  ı, each fut g0T 1  M0 ./ satisfying kut  vk  ı; t D 0; : : : ; T  1 and each ./-program fzt gTtD0 which satisfies .z0 ; xN v /  rNv =2; h.zT / C

T 1 X t D0

1 ut .zt ; zt C1 /  .h; fut gtTD0 ; 0; T; z0 /  ı

Q there exists fxt g1 N such that t D0 2 P.g; v/ .zT t ; xt /  ; t D 0; : : : ; 0 : Assume that u 2 M ./; ku  vk  ı; h 2 Mu .X /; kh  gk  ı; Q N /: fxt g1 t D0 2 P.h; u

(106) (107)

In view of (106), (107) and (A3), lim xt D xN u :

t !1

(108)

It follows from (100), (106) and the choice of ı0 that (99) holds. By (108), there exists an integer S0 > T0 such that .xS0 ; xN u /  rNv =4: Together with (99) this implies that .xS0 ; xN v /  rNv =2:

(109)

zt D xS0 t ; t D 0; : : : ; S0 :

(110)

Set

126

A.J. Zaslavski

By (109) and (110), .z0 ; xN v /  rNv =2:

(111)

0 is an ./-program. We show that It is clear that fzt gSt D0

h.zS0 / C

SX 0 1 t D0

u.zt ; zt C1 / D .h; u; S0 ; z0 /:

(112)

0 Let fyt gSt D0 be an ./-program satisfying

y0 D z0 :

(113)

In order to prove (112) it is sufficient to show that SX 0 1

h.zS0 / C

t D0

u.zt ; zt C1 /  h.yS0 / C

SX 0 1

u.yt ; yt C1 /:

t D0

It follows from (110) that h.zS0 / C

SX 0 1 t D0

u.zt ; zt C1 / D h.x0 / C

SX 0 1 t D0

uN .xS0 t 1 ; xS0 t /:

(114)

Set yNt D yS0 t ; t D 0; : : : ; S0 :

(115)

In view of (115), h.yS0 / C

SX 0 1 t D0

u.yt ; yt C1 / D h.yN0 / C

SX 0 1 t D0

uN .yNt ; yNt C1 /:

(116)

N In view of (107), (109), (110), (113)–(116), Proposition 4, Corollary 1 and .Nu; /overtaking optimality of fxt g1 (see (107)), t D0 h.zS0 / C

SX 0 1 t D0

D h.x0 / C

u.zt ; zt C1 /  Œh.yS0 / C

SX 0 1 t D0

SX 0 1

uN .xt ; xt C1 /  Œh.yN0 / C

u.yt ; yt C1 /

t D0

SX 0 1 t D0

uN .yNt ; yNt C1 /

Discrete Time Optimal Control Problems on Large Intervals

D h.x0 / C

.h.yN0 / C

SX 0 1

ŒNu.xt ; xt C1 /  u.xN u ; xN u /   uN .xt / C  uN .xt C1 /

t D0

SX 0 1 t D0

127

C uN .x0 /   uN .xS0 /

ŒNu.yNt ; yNt C1 /  u.xN u ; xN u /   uN .yNt / C  uN .yNt C1 / C uN .yN0 /   uN .yNS0 //

D h.x0 / C  uN .x0 /  h.yN0 /   uN .yN0 / 

SX 0 1 t D0

ŒNu.yNt ; yNt C1 /  u.xN u ; xN u /   uN .yNt / C  uN .yNt C1 /

 h.x0 / C  uN .x0 /  h.yN0 /   uN .yN0 /  0:

Thus (112) holds. By (106), (111), (112), the inequality S0 > T0 and the property (ii) 0 Q applied to fzt gSt D0 there exists an fxt g1 N such that for all t D 0; : : : ; 0 , t D0 2 P.g; v/   .zS0 t ; xt / D .xt ; xt /: Proposition 19 is proved.

10 Proof of Theorem 17 In the sequel we use the following auxiliary results. Proposition 20 (Proposition 12.1 of [39]). Let v 2 M ./, fxt g1 t D0 be a .v; /overtaking optimal and .v; /-good program and t2 > t1 be nonnegative integers such that xt1 D xt2 . Then xt D xN v for all integers t D t1 ; : : : ; t2 . Proposition 21 (Proposition 12.2 of [39]). Let v 2 M ./, fxt g1 t D0 be a .v; /overtaking optimal program such that x0 D xN v . Then xt D xN v for all integers t  0. For any .x; y/ 2 X  X and any nonempty set D  X  X put ..x; y/; D/ D inff.x; z1 / C .y; z2 / W .z1 ; z2 / 2 Dg: Since the mapping v ! v, N v 2 M ./ is an isometry Theorem 17 follows from Propositions 14, 15 and the following result. Proposition 22. Let M be either MN0 ./ or MNc;0 ./ and g 2 Mu .X /. Then there exists a set F  M \ M ./ which is a countable intersection of open everywhere dense subsets of M such that for each v 2 F there exists a unique point zv 2 X

128

A.J. Zaslavski

such that . v C g/.zv / D sup. v C g/ v and there exist a unique .v; /-overtaking optimal program fzvt g1 t D0 satisfying z0 D zv .

Let M be either MN0 ./ or MNc;0 ./. By Theorem 8, there exists a set F0  M \ M ./ which is a countable intersection of open everywhere dense subsets of M. For each g 2 Mu .X / denote by Eg the set of all v 2 M \ M ./ for which the following property holds: (Pi)

There exists a unique point zv 2 X such that . v C g/.zv / D sup. v C g/

and there exists a unique .v; /-overtaking optimal program fzvt g1 t D0 satisfying zv0 D zv . We proceed the proof of Proposition 22 with the following auxiliary result. Lemma 4. For each g 2 Mu .X /, Eg is a an everywhere dense subset of M. Proof. Let v 2 M \ M ./. It is sufficient to show that for any neighborhood U of v in M, U \ Eg 6D ;. There are two cases: . v C g/.xN v / D sup. v C g/I

(117)

. C g/.xN v / < sup. C g/:

(118)

v

v

Assume that (117) holds. Let 2 .0; 1/. Define v .x; y/ D v.x; y/  ..x; xN v / C .y; xN v //; .x; y/ 2 :

(119)

It is easy to see that v 2 M \ M0 ./ with xN v D xN v :

(120)

In view of (119), every .v ; /-good program fxt g1 t D0 is .v; /-good and lim xt D xN v :

t !1

Hence v 2 M ./:

(121)

Discrete Time Optimal Control Problems on Large Intervals

129

Proposition 6 and (119)–(121) imply that v .xN v ; xN v / D v.xN v ; xN v /;  v .y/   v .y/; y 2 X;  v .xN v / D  v .xN v / D 0;

. v C g/.y/  . v C g/.y/; y 2 X; . v C g/.xN v / D . v C g/.xN v /:

(122)

Assume that a point z 2 X satisfies . v C g/.z/ D sup. v C g/

(123)

and that fzt g1 t D0 is an .v ; /-overtaking optimal program such that z0 D z:

(124)

In view of (117), (119)–(124) and Proposition 8, sup. v C g/  g.z/ D sup. v C g/  g.z/ D  v .z/ D lim

T !1

T 1 X

D lim ΠT !1

t D0

 lim sup

t D0

Œv .zt ; zt C1 /  v .xN v ; xN v /

v.zt ; zt C1 /  T v.xN v ; xN v / 

T 1 X

T !1 t D0

T 1 X

.v.zt ; zt C1 /  v.xN v ; xN v //    v .z/ 

D 

T 1 X

1 X t D0

t D0

..zt ; xN v / C .zt C1 ; xN v //

1 X ..zt ; xN v / C .zt C1 ; xN v // t D0

..zt ; xN v / C .zt C1 ; xN v //

1 X ..zt ; xN v / C .zt C1 ; xN v // C . v C g/.z/  g.z/: t D0

This implies that zt D xN v for all integers t  0 and v 2 Eg . Assume that (118) holds. There exist a point z 2 X which satisfies . v C g/.z / D sup. v C g/ and an .v; /-overtaking optimal program fzt g1 t D0 satisfying z0 D z : By Proposition 8 and (125),

(125)

130

A.J. Zaslavski

 v .z / D lim

T !1

T 1 X t D0

Œv.zt ; zt C1 /  v.xN v ; xN v /:

(126)

Assumption (A3) imply that lim z t !1 t

D xN v :

(127)

By (118), (125), (127) and Proposition 10, . v C g/.zt / < . v C g/.z0 / for all large enough integers t  1:

(128)

In view of (128), there exists an integer 0  0 such that . v C g/.z0 / D . v C g/.z / and that . v C g/.zt / < . v C g/.z / for all integers t > 0 . We may assume without loss of generality 0 D 0. Thus . v C g/.z0 / D sup. v C g/; . v C g/.zt / < . v C g/.z0 / for all integers t  1: (129) Let 2 .0; 1/. For all .x; y/ 2  define v .x; y/ D v.x; y/  ..x; y/; .f.zt ; zt C1 / W t D 0; 1; : : : g [ f.xN v ; xN v /g//: (130) By (130), v 2 M \ M0 ./; v .xN v ; xN v / D v.xN v ; xN v /; every .v ; /-good program is .v; /-good and converges to xN v . Hence v 2 M ./; xN v D xN v :

(131)

It follows from (125), (126), (130), (131) and the equality z0 D z that  v .y/   v .y/; y 2 X;  v .z / D  v .z /; sup. v C g/ D sup. v C g/:

(132)

By (131), for all natural numbers T , .v ; T; xN v ; xN v / D T v.xN v ; xN v /:

(133)

Discrete Time Optimal Control Problems on Large Intervals

131

Proposition 6 and (131) imply that  v .xN v / D 0:

(134)

K D f.zt ; zt C1 / W t D 0; 1; : : : g [ f.xN v ; xN v /g:

(135)

Set

Assume that y 2 X satisfies . v C g/.y/ D sup. v C g/

(136)

and that fyt g1 t D0 is an .v ; /-overtaking optimal program satisfying y0 D y:

(137)

By (125), (130)–(132), (135)–(137), Proposition 8 and since the program fyt g1 t D0 is .v ; /-overtaking optimal we have sup. v C g/ D . v C g/.z / D . v C g/.z / D . v C g/.y/ D lim

T !1

T 1 X

D lim ΠT !1

t D0

T 1 X t D0

Œv .yt ; yt C1 /  v .xN v ; xN v / C g.y/

.v.yt ; yt C1 /  v.xN v ; xN v /  ..yt ; yt C1 /; K// C g.y/

  v .y/ 

1 X t D0

..yt ; yt C1 /; K/ C g.y/:

Together with (118) and (133) this implies that . v C g/.y/ D . v C g/.z /;

(138)

.yt ; yt C1 / 2 K for all integers t  0:

(139)

In view of (118), (125), (129), (135), (137)–(139) and the equality z0 D z , y D y0 D z :

(140)

We show by induction that yt D zt for all integers t  0. There are two cases: zt 6D xN v for all integers t  0I xN v 2 fzt W t D 0; 1; : : : g:

(141) (142)

132

A.J. Zaslavski

Assume that (141) holds. By Proposition 20, (125)–(126) and (141), zt1 6D zt2 for all integers t2 > t1  0:

(143)

Assume that T  0 is an integer and that yt D zt ; t D 0; : : : ; T:

(144)

(Note that in view of (140) and the equality z0 D z our assumption holds for T D 0.) By (135), (139), (141), (143) and (144), .zT ; yT C1 / D .yT ; yT C1 / 2 K D f.zt ; zt C1 / W t D 0; 1; : : : g [ f.xN v ; xN v /g and yT C1 D zT C1 . Thus yt D zt for all integers t  0. Assume that (142) holds. By (118), (125) and the equality z0 D z , there is a natural number S such that zS D xN v ; zt 6D xN v for all integers t 2 Œ0; S /:

(145)

Propositions 20 and 21 imply that,

zt2

6D

zt1

zt D xN v for all integers t  S;

(146)

for all integers t1 ; t2 2 Œ0; S  such that t1 < t2 :

(147)

Assume that T  0 is an integer and that yt D zt ; t D 0; : : : ; T:

(148)

(Note that in view of (140), our assumption holds for T D 0.) If T < S , then by (135), (139), (145) and (148), .zT ; yT C1 / D .yT ; yT C1 / 2 f.zt ; zt C1 / W t D 0; 1; : : : g [ f.xN v ; xN v /g and yT C1 D zT C1 . If T  S , then by (135), (139), (145), (146) and (148), .xN v ; yT C1 / D .yT ; yT C1 / 2 f.zt ; zt C1 / W t D 0; 1; : : : g [ f.xN v ; xN v /g and yT C1 D xN v D zT C1 . Thus yt D zt for all integers t  0 in the both cases (see (141) and (142)). This implies that v 2 Eg . Therefore the inclusion above holds in the both cases (see (117) and (118)). Since v ! v as ! 0C in M we conclude that for any neighborhood U of v in M, U \ Eg 6D ;: Thus Eg is an everywhere dense subset of M. Lemma 4 is proved. Completion of the proof of Proposition 22 By definition, for every v 2 Eg , there exist a unique .v; /-overtaking optimal program fzvt g1 t D0 satisfying

Discrete Time Optimal Control Problems on Large Intervals

133

. v C g/.zv0 / D sup. v C g/: Let v 2 Eg and k  1 be an integer. By Proposition 17, there exist an open neighborhood U .v; k/ of v in M and an integer T .v; k/  k such that the following property holds: Q (Pii) For each u 2 U .v; k/ \ M ./ and each fxt g1 t D0 2 P.g; u/ we have .xt ; zvt /  k 1 , t D 0; : : : ; k. Set F1 D \1 pD1 [ fU .v; k/ W v 2 Eg ; k  pg; F D F1 \ F0 :

(149)

Clearly, F is a countable intersection of open everywhere dense subsets of M and F  F0  M ./. .i / Let u 2 F , p  1 be an integer and fxt g1 t D0 , i D 1; 2 be .u; /-overtaking optimal programs such that .i /

. v C g/.x0 / D sup. v C g/; i D 1; 2:

(150)

By (149), there exist vp 2 Eg and an integer kp  p such that u 2 U .vp ; kp /: .i /

(151) v

In view of (149), (151) and property (Pii), .xt ; zt p /  kp1  p 1 , t D 0; : : : ; p, .1/

.2/

i D 1; 2. This implies that .xt ; xt /  2p 1 , t D 0; : : : ; p. Since p is any .2/ .1/ natural number we conclude that xt D xt for all integers t  0: Proposition 22 is proved. t u

11 Proof of Proposition 18 By Theorem 8, there exists a set G0  M\M ./ which is a countable intersection of open everywhere dense subsets of M. Denote by E the set of all .v; g/ 2 .M \ M .//  A for which there exists a .v;g/ Q unique program fzt g1 t D0 2 P.g; v/. By Lemma 4, E is a an everywhere dense subset of M  A. Let .v; g/ 2 E and k  1 be an integer. By Proposition 17, there exist an open neighborhood U .v; g; k/ of .v; g/ in M  A and an integer T .v; g; k/  k such that the following property holds: Q (i) For each .u; h/ 2 U .v; g; k/ \ .M ./  A/ and each fxt g1 t D0 2 P.h; u/ we .v;g/ 1 have .xt ; zt /  k , t D 0; : : : ; k.

134

A.J. Zaslavski

Set F D \1 pD1 [ fU .v; g; k/ W .v; g/ 2 E; k  pg \ .G0  A/:

(152)

Clearly, F is a countable intersection of open everywhere dense subsets of M  A. Let .u; h/ 2 F , p  1 be an integer and .i / Q fxt g1 t D0 2 P.h; u/; i D 1; 2:

(153)

By (152), there exist .vp ; gp / 2 E and an integer kp  p such that .u; h/ 2 U .vp ; gp ; kp /: .i /

(154)

.gp ;vp /

/  kp1  p 1 , t D 0; : : : ; p, i D 1; 2. This .2/ .1/ implies that .xt ; xt /  2p 1 , t D 0; : : : ; p. Since p is any natural number we .1/ .2/ conclude that xt D xt for all integers t  0: Proposition 18 is proved. t u In view of (152)–(154), .xt ; zt

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Index

A Asymptotic turnpike property, 93 Aumann-Gelfand integral, X over set A, 4 Aumann integral, 9, 10 B Baire category, 93 Banach space, 100 Bermudan derivatives, 58, 59 Bolza problem, 92 C Cardinality, 96 cc (E), 3 ck (E), 3 Closed convergence topology, 34, 35, 39, 41, 45 Compact metric space, 92 Conditional expectation, level sets, 11, 12, 13, 14, 19 cwk (E), 3 D Demand function, 34–37, 41, 43–45 Discrete-time optimal control system, 92 E Estimate error, 34, 35, 41, 45 Expectation of random fuzzy convex integrands, 18

Expectation of the level sets, 11, 12 Expected value (or expectation), 26

F F-measurable, 3 Fuzzy conditional expectation, 18, 19, 26 Fuzzy convex upper semicontinuous variable, 8 Fuzzy expectation, 18, 19 Fuzzy martingale, 26 Fuzzy set valued random variables, 2 Fuzzy submartingale, 26 Fuzzy supermartingale, 26

G Gelfandintegrable selections, 4 (/-Good program, 93

H Hörmander type diffusion process, 61, 78–81

I Infinite horizon optimal control problem, 93 Infinite interval, 92 Integrability theory, 3, 34 Integrably bounded mappings, 4 Integrand martingales, 27 Inverse demand function, 35, 36, 38, 41, 45, 52

137

138 K K -valued random set, 5 L Lagrange problem, 92 Least square regression methods, 58 Local C 1 topology, 35, 41–43, 45 Local uniform topology, 41, 45 Lower semicontinuous integrand martingale, 27 M Martingales, 27 Monte Carlo methods, 58 N Normal integrand, 4

Index R Random fuzzy convex upper semicontinuous integrands, level sets, 9, 12 Random fuzzy convex upper semicontinuous variable, level sets, 8–9 Random lower semicontinuous integrands, 8 Random systems of piece-wise polynomials, 60 Random upper semicontinuous integrands, 8 Re-simulation, 71–73 Revealed preference theory, 34

S Scalarly F-measurable mapping, 3, 4 Scalarly integrable, 4 Sequential weak lower limit w -liXn , 6 Sequential weak upper limit w -lsXn , 6 Support function, 3

O Optimal control problem, 92 Overtaking optimal program, 98, 99

T Turnpike phenomenon, 93

P Perturbation, 98 Preference relation, 34, 35, 38, 42, 43, 45 (/-Program, 92 p-transitive, 34, 36, 38, 42, 43

W Weak axiom of revealed preference, 34 Weakly compactly generated (WCG), 6 Weak star (w K for short) converges, 6 w -topology, 3