Non-Gaussian Selfsimilar Stochastic Processes 3031337719, 9783031337710

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SpringerBriefs in Probability and Mathematical Statistics Ciprian Tudor

Non-Gaussian Selfsimilar Stochastic Processes

SpringerBriefs in Probability and Mathematical Statistics Series Editors Nina Gantert, Technische Universität München, Munich, Germany Tailen Hsing, University of Michigan, Ann Arbor, MI, USA Richard Nickl, University of Cambridge, Cambridge, UK Sandrine Péché, Univirsité Paris Diderot, Paris, France Yosef Rinott, Hebrew University of Jerusalem, Jerusalem, Israel Almut E.D. Veraart, Imperial College London, London, UK Editor-in-Chief Gesine Reinert, University of Oxford, Oxford, UK

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Ciprian Tudor

Non-Gaussian Selfsimilar Stochastic Processes

Ciprian Tudor Département de Mathématiques Université de Lille Villeneuve-d’Ascq, France

ISSN 2365-4333 ISSN 2365-4341 (electronic) SpringerBriefs in Probability and Mathematical Statistics ISBN 978-3-031-33771-0 ISBN 978-3-031-33772-7 (eBook) https://doi.org/10.1007/978-3-031-33772-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The origin of this short monograph lies in the lectures I gave at the Finnish Summer School on Probability and Statistics, held in Lammi, Finland, in May 2022. It concerns a particular class of self-similar stochastic processes, the so-called Hermite processes. Self-similar processes are stochastic processes that are invariant in distribution under a suitable time scaling. The most known self-similar process is the fractional Brownian motion (fBm), which can be defined as the only Gaussian self-similar process with stationary increments. Its stochastic analysis constitutes an important research direction in probability theory nowadays. The Hermite processes are nonGaussian extensions of the fBm. These processes, which are also self-similar, with stationary increments and exhibit long-range dependence, have been also intensively studied in the last decades. The purpose is to offer a rather detailed description of this class of stochastic processes and to discuss some recent developments concerning their stochastic and statistical analysis. We analyze some stochastic (partial) differential equations driven by a Hermite process, as well as the estimation of certain parameters in stochastic models with Hermite noise. The manuscript starts with an introduction to Wiener chaos and multiple stochastic integrals Chap. 1. This first part is motivated by the fact that almost all the random variables and processes appearing throughout this manuscript can be expressed as finite sums of multiple stochastic integrals. The main part of the material included in this introductory chapter is already contained in some references on the analysis on Wiener space or Malliavin calculus, such as [26 or 27]. In Chaps. 2–4 we introduce the Hermite processes and their multiparameter version, the Hermite sheets. We discuss their main properties, as well as the basic aspects of the stochastic integration with respect to them. We also included an analysis of some stochastic (partial) differential equations driven by an Hermite noise. In particular, we present a detailed study of the Hermite Ornstein-Uhlenbeck process (the solution of the Langevin equation with Hermite noise) and of the solution to the stochastic heat equation driven by such a random perturbation. The last part of this book (Chap. 5) concerns the parameter estimation for Hermiterelated models. While the statistical inference for stochastic equation driven by the Brownian motion, or more general, by a Gaussian noise, has a long history, the v

vi

Preface

statistical inference for systems driven by Hermite processes and sheets is only at its beginning. We here present some techniques to estimate certain parameters in stochastic models involving a Hermite noise. Although we assume that the reader has some background on the basics of probability theory and stochastic process, our intention to keep the book self-contained, as much as possible. Villeneuve-d’Ascq, France

Ciprian Tudor

About This Book

The stochastic processes treated in this short book are mainly characterized by the following facts: they are self-similar with stationary increments and they belong to a Wiener chaos of fixed order. The self-similarity (or the scaling property) of a object roughly means that a part of it resembles to the whole object. This property can be observed in nature or in the real life in many situations. For example, the coastlines, the top of trees or the internet traffic are approximately self-similar. For a stochastic process (X t , t ≥ 0), the selfsimilarity can be described by the following: there exists a real-numer H ∈ (0, 1) such  H that for every real number c > 0, the stochastic processes (X ct , t ≥ 0) and c X t , t ≥ 0 have the same finite-dimensional distributions. We say that the process X is H -self-similar. A stochastic process (X t , t ≥ 0) has stationary increments if the statistical characteristics of its increments of fixed lenght do not vary over time. This property is also observed in pratice for physical phenomena driven by a stationary source. Mathematically speaking, the process (X t , t ≥ 0) has stationary increments if for every h > 0, the finite-dimensional distributions of the stochastic processes (X t+h − X h , t ≥ 0) and (X t , t ≥ 0) are the same. The most proeminent example of a self-similar  process with stationary  stochastic increments is the fractional Brownian motion BtH , t ≥ 0 . It is defined as a zeromean Gaussian process with covariance function R(t, s) = EBtH BsH =

 1  2H t + s 2H − |t − s|2H , 2

(1)

for any s, t ≥ 0. The index H is assumed to be in the interval (0, 1) and it is called the Hurst parameter, or the Hurst index. This parameter characterizes the main properties of the fractional Brownian motion (fBm) such as the regularity of the sample paths or the scaling property (the fBm is H -self-similar). If H = 21 , the fBm is nothing else than the well-known standard Brownian motion. If a stochastic process (X t , t ≥ 0) is H -self-similar and it has stationary increments, then the exact formula of its covariance can be obtained. If we assume that X is

vii

viii

About This Book

centered and X 1 has unit variance, then the covariance of the process X is given by the right-hand side of (1). Since the expectation and the covariance function determine the law of a Gaussian process, it follows that there exists (modulo a multiplicative constant) only one Gaussian self-similar process with stationary increments which is the fractional Brownian motion. There exists a vast (recent or less recent) literature on fBm and our goal is not the focus on the analysis of this particular stochastic process. The purpose here is analyze a larger class of self-similar processes with stationary increments, the socalled Hermite processes. The class of Hermite processes includes the fractional Brownian motion but it mainly contains non-Gaussian processes. A particularity of the Hermite processes is that they live in a Wiener chaos of fixed order. The Wiener chaos can be defined with respect to any isonormal Gaussian process defined on a probability space (Ω, F, P) , in particular with respect to the Brownian motion. The first Wiener chaos (i.e. the Wiener chaos of order one) contains only Gaussian random variables while the elements of the other chaoses have nonGaussian distribution. Two Wiener chaoses of different orders are orthogonal with respect with respect to the scalar product in L 2 (Ω). The Wiener chaos expansion constitutes a fundamental result in stochastic analysis. This result states that any square integrable random variable F (mesurable with respect to the sigma-algebra general by the underlying Gaussian process) can be expressed as an orthogonal infinite sum of Wiener chaoses, i.e.  Fn F= n≥0

where F0 = EF and for each n ≥ 1, the term Fn belongs to the nth Wiener chaos, denoted Hn . In the case when the underlying Gaussian process is the standard Brownian motion, each random variable Fn can be written as a nth iterated Itô integral with respect to the Brownian motion, which is also called multiple stochastic integral (or multiple Wiener-Itô integral) of order  n ≥ 1.  H,q The Hermite process Z t , t ≥ 0 is characterized by its order q ≥ 1 (q is an   integer number) and its self-similarity index, a real number H ∈ 21 , 1 . The Hermite H,q process of order q belongs to the qth Wiener chaos (in the sense that Z t belongs to Hq for every t ≥ 0). When q = 1, the corresponding Hermite process is the fractional Brownian motion, which is the only Gaussian Hermite process. The random variable H,q Z t can be then written as a multiple stochastic integral of order q with respect to the Wiener process. Therefore, we start our book with a detailed presentation of the Wiener chaos and of the multiple stochastic integrals. After this first chapter dedicated to the construction and the properties of the multiple stochastic integrals and Wiener chaos, the rest of the material is consecrated to the analysis of the Hermite processes. We survey the findings on the distributional and trajectorial properties of this processes obtained in the past decades. As a general matter of fact, the stochastic analysis of this class of stochastic processes is challenging due to the following facts: their probability distribution is pretty complex

About This Book

ix

and they are not semi-martingales. One of the purposes is to analyze the law of this processes. Some properties of this law can be obtained from its integral representation (such as the scaling property, the moments or the stationarity of the increments). As a consequence of these properties, we can also derive some of the trajectorial properties of the Hermite processes (the regularity of the sample paths or the behavior of its p-variation). But the general characterization of the distribution of a Hermite process is not known. While the covariance function of a Hermite process, of any order q ≥ 1, il always given by (1), this covariance determines the distribution of the process only for q = 1 (the Gaussian case). In the case q = 2, the corresponding Hermite process is also called the Rosenblatt process. It was introduced in 35 and then it was made more popular by the works [16, 40] or more recently [45]. This stochastic process will receive a particular attention in our notes, because it belongs to the second Wiener chaos and in this case we know some facts about its probability distribution (which is characterized by its cumulants, or equivalently, by its moments). For a Hermite process of order q ≥ 3, we have little knowledge about its probability law. Besides the analysis of the basic properties of the class of Hermite processes, another goal of this manuscript is to develop a stochastic analysis with respect to these stochastic processes. As mentioned above, the main difficulty comes from the fact that the Hermite processes are not semi-martingales (they are actually zero quadratic variation processes). Therefore the classical Itô calculus cannot be applied to them. A stochastic integration theory is nowadays well-developed for the fractional Brownian motion by using various approaches (Malliavin calculus, rough paths theory, stochastic calculus via regularization etc) but these methods cannot be directly applied to the Hermite processes of order q ≥ 2 due to their nonGaussian character. Although there are some (pretty technical) attempts to define a stochastic integral of random integrands with respect to the Hermite processes, we will restrict here to the case of deterministic integrands, by presenting the construction and the basic properties of the Wiener integral with respect to the Hermite process. A related question considered in the manuscript is the analysis of the so-called Hermite Ornstein-Uhlenbeck, defined as the solution to the Langevin equation with Hermite noise. We will also introduce a multiparameter version of the Hermite process, i.e. a stochastic field   H,q,d Zt , t ∈ Rd . This random field is called the multiparameter Hermite process, or the Hermite sheet. We will discuss various aspects of it. This random field is also self-similar with stationary increments but these concepts has now to be understood in a multidimensional context. After a survey of the main properties of the multiparameter stochastic processes, we introduce a Wiener integral with respect to them. This allows to consider stochastic partial differential equations (SPDEs) with Hermite noise. We here illustrate the case of the stochastic heat equation (although other types of SPDEs, such as the stochastic wave equation, have been treated in the literature). We study

x

About This Book

when the mild solution is well-defined and we analyze the distribution and sample paths of this solution. The last section is devoted to some applications of our theoretical results to statistics. Although the literature on statistical inference for Hermite-driven models is not yet very vast, we present two situations when various parameters appearing in such models can be estimated. These two situations concern the case of the Langevin equation with Hermite noise and of the stochastic heat equation driven by an additive Hermite sheet. In these examples, the estimation technique is based on the p-variation of the observed process, which constitutes a standars method for parameter estimation. For a given stochastic process(X t , t ≥ 0), its p-variation is usually defined as N −1   Xt

i+1

p − X ti 

i=0

for i = 1, . . . , N . In terms of this p-variation, for some p > 0, where ti = we define estimators for the parameters that appear in our model. By analyzing the behavior of the above sequence, we derive the asymptotic properties of the associated estimators. i N

Contents

1 Multiple Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Isonormal Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The Wiener Integral with Respect to the Wiener Process . . . . 1.1.2 The Wiener Integral with Respect to the Brownian Sheet . . . 1.2 Multiple Wiener-Itô Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 A First Product Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 The Wiener Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 The General Product Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Random Variables in the Second Wiener Chaos . . . . . . . . . . . . . . . . . .

1 1 2 5 6 6 11 16 20 21

2 Hermite Processes: Definition and Basic Properties . . . . . . . . . . . . . . . . . 2.1 The Kernel of the Hermite Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Definition of the Hermite Process and Some Immediate Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Self-similarity and Stationarity of the Increments . . . . . . . . . . 2.2.2 Moments and Hölder Continuity . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 The Hermite Noise and the Long Memory . . . . . . . . . . . . . . . . 2.2.4 p-Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Approximation by Semimartingales . . . . . . . . . . . . . . . . . . . . . . 2.3 Some Particular Hermite Processes: Fractional Brownian Motion and the Rosenblatt Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Fractional Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The Rosenblatt Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Alternative Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 On the Simulation of the Rosenblatt Process . . . . . . . . . . . . . . . . . . . . .

25 27

3 The Wiener Integral with Respect to the Hermite Process and the Hermite Ornstein-Uhlenbeck Process . . . . . . . . . . . . . . . . . . . . . . 3.1 Wiener Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Cases q = 1 and q = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Wiener Integral in the Riemann-Stieltjes Sense . . . . . . . . . . . . . . . . . .

29 30 32 32 33 34 36 36 36 38 43 45 46 49 51 xi

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Contents

3.4 The Hermite Ornstein-Uhlenbeck Process . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.1 Definition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.2 The Stationary Hermite Ornstein-Uhlenbeck Process . . . . . . . 56 4 Hermite Sheets and SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Definition of the Hermite Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Wiener Integral with Respect to the Hermite Sheet . . . . . . . . . . . . . . . 4.4 The Stochastic Heat Equation with Hermite Noise . . . . . . . . . . . . . . . . 4.4.1 Existence of the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Regularity of Sample Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 A Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 p-Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Statistical Inference for Stochastic (Partial) Differential Equations with Hermite Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Parameter Identification for the Hermite Ornstein-Uhlenbeck Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Quadratic Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Estimation of the Hurst Parameter . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Estimation of σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Drift Estimation for the Stochastic Heat Equation with Hermite Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 60 63 65 69 70 73 75 78 80 85 86 87 92 94 95

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Chapter 1

Multiple Stochastic Integrals

The random variables and the stochastic processes discussed below will in general be defined as multiple stochastic integrals (or multiple Wiener-Itô integrals) and they belong to a Wiener chaos. Therefore, we start our manuscript with a preliminary chapter where we included the main properties of the Wiener chaos. The Wiener chaos is constructed with respect to the so-called isonormal Gaussian process and we start with a presentation of this concept. Then the rest of the chapter focuses on the definition and on the main properties of the Wiener chaos and of the multiple stochastic integrals.

1.1 Isonormal Processes The isonormal processes are families of Gaussian random variables indexed by an Hilbert space. They are the basis of the construction of the Wiener chaos. We will describe them, by focusing on the examples which we are going to use in these lectures: the Brownian motion and the Brownian sheet. Let (., F , P) be a probability space. Let (H, H ) be a real and separable Hilbert space. Let || · || H denote the norm in H . Definition 1.1 A centered Gaussian family (W (h), h ∈ H ) on (., F ., P) such that for every h 1 , h 2 ∈ H , (1.1) EW (h 1 )W (h 2 ) = H is called an isonormal process. It follows from Definition 1.1 that for every h ∈ H , the random variable W (h) satisfies ) ( (1.2) W (h) ∼ N 0, ||h||2 .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Tudor, Non-Gaussian Selfsimilar Stochastic Processes, SpringerBriefs in Probability and Mathematical Statistics, https://doi.org/10.1007/978-3-031-33772-7_1

1

2

1 Multiple Stochastic Integrals

It can also be shown that the mapping h ∈ H → W (h) ∈ L 2 (.) is linear. A particular case is when H = L 2 (T , B, μ), where T non-empty, B is a sigmaalgebra included in B(T ), and μ is a sigma-finite measure without atoms. In this case, we will use the notation W ( A) := W (1 A ), for A ∈ Bb

(1.3)

where we denoted by Bb the set of A ∈ B with μ( A) < ∞. The family (W (A), A ∈ Bb ) satisfies the following properties: • For every A ∈ Bb , from (1.2), W (A) ∼ N (0, μ( A)). • For every A ∈ Bb , from (1.1), EW ( A)W (B) = μ(A ∩ B).

(1.4)

• For every A ∈ Bb , the random variables W (A) and W (B) are independent if and only if μ(A ∩ B) = 0, as a consequence of (1.4) and of the fact that the family (W (A), A ∈ Bb ) is Gaussian. An easy way to construct an isonormal process in an arbitrary real and separable Hilbert space H is as follows. Let (e j , j ≥ 1) be an orthonormal complete system in H and let (X j , j ≥ 1) be a family of independent real-valued standard normal random variables. For every h ∈ H , set W (h) =

E

H X j .

j≥1

Then (W (h), h ∈ H ) is a Gaussian family of centered random variables and for every h 1 , h 2 ∈ H , we have EW (h 1 )W (h 2 ) = H . So (W (h), h ∈ H ) is an isonormal process. Other useful examples of isonormal processes are given below.

1.1.1 The Wiener Integral with Respect to the Wiener Process The two-sided Brownian motion is a centered Gaussian process (W (t), t ∈ T ⊂ R) with covariance given by EWt Ws =

1 (|t| + |s| − |t − s|) , 2

s, t ∈ T .

(1.5)

1.1 Isonormal Processes

3

When T = [0, ∞), then (Wt , t ≥ 0) is called the standard Brownian motion. In this case, we have EWt Ws = t ∧ s, for every s, t ≥ 0. Remark 1.1 There is an alternative way to define the two-sided Wiener process. Consider two independent (standard) Brownian motions (Wt(1) , t ≥ 0) and (Wt(2) , t ≥ 0). We set (Wt , t ∈ R) by { Wt(1) for t ≥ 0 (1.6) Wt = (2) for t < 0. W−t Then for every s, t ∈ R, ⎧ ⎪ ⎨t ∧ s, if s, t ≥ 0 EWt Ws = 0 if t < 0, s ≥ 0 or s < 0, t ≥ 0 ⎪ ⎩ (−t) ∧ (−s) if s, t < 0

=

1 (|t| + |s| − |t − s|) . (1.7) 2

Hence (Wt , t ∈ R) is a centered Gaussian process with covariance (1.5). Let T ⊂ R and let (Wt , t ∈ T ) be a Wiener process. We can naturally associate to it an isonormal process indexed by the Hilbert space (L 2 (T ), B, λ) where B is a sigma-algebra included in B(T ) and λ is the Lebesque measure. Let a < b and A = [a, b]. We set W (A) = Wb − Wa . Let E1 be the set of functions h ∈ H = L 2 (T ) of the form h(t) =

N E

ci 1 Ai (t),

t ∈ T.

(1.8)

i=1

with N ≥ 1, ci ∈ R, and A1 , ..., An disjoint segments in Bb (the set of Borel subsets with finite Lebesgue measure in B). Then we set W (h) =

N E

ci W ( Ai ).

i=1

We notice that EW (h)2 =

N E i=1

ci2 EW (Ai )2 =

N E

ci2 λ( Ai ) = ||h||2L 2 (T ) .

(1.9)

i=1

Now, let h ∈ L 2 (T ). Since the set E1 is dense in H = L 2 (T ), there exists a sequence (h n , n ≥ 1) of simple functions of the form (1.8) such that ||h n − h|| L 2 (T ) →n→∞ 0.

4

1 Multiple Stochastic Integrals

We set W (h) = lim W (h n ) in L 2 (.). n→∞

(1.10)

It is standard to see that: • the above limit exists • the limit (1.10) does not depend on the chosen approximating sequence (h n , n ≥ 1). Indeed, (W (h n ), n ≥ 1) is a Cauchy sequence in L 2 (.) since for m, n ≥ 1, E (W (h m ) − W (h n ))2 = ||h m − h n ||2L 2 (T ) →m,n→∞ 0 since (h n , n ≥ 1) converges in L 2 (T ) (to h). Thus (W (h n ), n ≥ 1) converges in L 2 (.). Also, if we consider two sequences (h n , n ≥ 1) and (gn , n ≥ 1) such that ||h n − h|| L 2 (T ) →n→∞ 0 and ||gn − h|| L 2 (T ) →n→∞ 0, then ( ) E (W (h n ) − W (gn ))2 = ||h n − gn ||2L 2 (T ) ≤ 2 ||h n − h||2L 2 (T ) + ||g − gn ||2L 2 (T ) →n→∞ 0.

Then (W (h), h ∈ H = L 2 (T )) becomes an isonormal process. Indeed, for every h, g ∈ H = L 2 (T ), let (h n , n ≥ 1) and (gn , n ≥ 1) be two sequence of simple functions of the form (1.8) such that ||h n − h|| L 2 (T ) →n→∞ 0 and ||h n − h|| L 2 (T ) →n→∞ 0. Then | H − H | ≤ | H | + | H | ≤ ||h n − h|| H ||gn || H + ||h|| H ||gn − g|| H →n→∞ 0 and EW (h)W (g) = lim EW (h n )W (gn ) = lim H = H . n→∞

n→∞

We also use the notation { W (h) =

h(s)dWs . T

The random variable W (h) is called the Wiener integral of h with respect to the Wiener process.

1.1 Isonormal Processes

5

Remark 1.2 Conversely, to any isonormal process indexed by a Hilbert space of L 2 type, one can associate a Brownian motion. Let (W (h), h ∈ L 2 (R+ )) be an isonormal process. Set, for every t ≥ 0, ) ( Bt = W 1[0,t] . Then (Bt , t ≥ 0) is a centered Gaussian process, starting from zero and for every s, t ≥ 0, ) ( ) ( EBt Bs = EW 1[0,t] W 1[0,s] = L 2 (R+ ) = t ∧ s. Consequently, (Bt , t ≥ 0) is a standard Brownian motion.

1.1.2

The Wiener Integral with Respect to the Brownian Sheet

The Brownian sheet constitutes a multiparameter version of the process defined by (1.5). Definition 1.2 Let N ≥ 1. The N -parameter Brownian sheet is defined as a centered Gaussian process (W (x), x ∈ T ⊂ R N ) with covariance EW (x)W (y) =

N ( || 1 j=1

2

) (|x j | + |y j | − |x j − y j |) ,

(1.11)

if x = (x1 , ..., x N ), y = (y1 , ..., y N ) ∈ T ⊂ R N . If N = 1, then (1.11) reduces to (1.5). If T = R+N , then the process with covariance (1.11) will be called the standard Brownian sheet. So, in this situation, EW (x)W (y) =

N ||

(x j ∧ y j ).

j=1

We also introduce the notion of the high-order increment of a d- parameter process X = (X (x), x ∈ Rd ) on a rectangle [s, t] ⊂ Rd , s = (s1 , ..., sd ), t = (t1 , ..., td ), with s ≤ t. This increment is denoted by /\X [s,t] and it is given by /\X [s,t] =

E

E

(−1)d−

i ri

X s+r·(t−s) .

r∈{0,1}d

When d = 1 one obtains the /\X [s,t] = X t − X s

(1.12)

6

1 Multiple Stochastic Integrals

while for d = 2 one gets /\X [s,t] = X t1 ,t2 − X t1 ,s2 − X s1 ,t2 + X s1 ,s2 . We can associate an isonormal process with the Wiener sheet (W (x), x ∈ T ⊂ R N ). If A is a rectangle of the form A = [a, b] = [a1 , b1 ] × [a2 , b2 ] × . . . × [a N , b N ], then we set W (A) = /\W[a,b] . The definition can be then extended to simple functions of the form h(x) =

M E

ci 1 Ai (x), x ∈ R N ,

i=1

where M ≥ 1, ci ∈ R and Ai are disjoint rectangles in Bb (T ), for i = 1, ..., M, by setting M E W (h) = ci W (Ai ). i=1

By the density of the simple functions in L 2 (T ), we can extend W (h), for every h ∈ L 2 (R N ) by (1.10). By following the one-parameter case, we can show that the family (W (h), h ∈ L 2 (T ) becomes an isonormal process, in the sense of Definition 1.1, i.e. it is a centered Gaussian process such that { EW (h 1 )W (h 2 ) = L 2 (R N ) =

RN

h 1 (x1 , ..., x N )h 2 (x1 , ..., x N )d x1 ...d x N .

{

We also use the notation W (h) =

TN

h(x)dW (x)

and we will call W (h) the Wiener integral of h with respect to the Wiener sheet W .

1.2 Multiple Wiener-Itô Integrals 1.2.1 Definition and Basic Properties We will present the construction of the Wiener chaos and of the multiple stochastic integrals on a Hilbert space of L 2 -type, although the construction of these objects can be done over a general underlying Hilbert space (see e.g. [27] or [29]). Let H = L 2 (T , B, μ) and let (W (h), h ∈ H ) an isonormal process. We denote by En the

1.2 Multiple Wiener-Itô Integrals

7

set of elementary functions of n variables f : T n → R. We say that f ∈ En if f can be written as N E

f (t1 , . . . , tn ) =

ai1 ,...,in 1 Ai1 ×Ai2 ×...×Ain (t1 , ..., tn ),

(1.13)

i 1 ,..,i n =1

where N ≥ 1, the set A1 , ..., A N ∈ Bb are disjoint and ai1 ,..,in = 0 if two indices are equal (i.e. if there exist k /= l with i k = il ). For instance f (t1 , t2 ) = 21[0,1)×[1,2) (t1 , t2 ) belongs to E2 but f (t1 , t2 , t3 ) = 21[0,1)×[1,2)×[0,1) (t1 , t2 , t3 ) does not belong to E3 . We first define the multiple stochastic integral of an elementary function. Recall the notation (1.3). Definition 1.3 For f : T n → R of the form (1.13), we set N E

In ( f ) =

ai1 ,...,in W (Ai1 ) . . . W ( Ain )

(1.14)

i 1 ,..,i n =1

We will call In ( f ) the multiple stochastic integral of order n of f with respect to isonormal process W . Notice that In ( f ) belongs to L 2 (.). If f : T m → R, we denote by ~ f its symmetrization, i.e. 1 E f˜(t1 , .., tm ) = f (tσ1 , .., tσm ). n! σ∈S

(1.15)

n

We have the following lemma. Lemma 1.1 For every f ∈ L 2 (T n ), || f˜|| L 2 (T n ) ≤ || f || L 2 (T n ) . Proof We have, by (1.15) and Cauchy-Schwarz, || f˜|| L 2 (T n )

1 = 2 n!

⎛

{

⎝ Tn

E

⎞2 f (tσ(1) , ..., tσ(n) )⎠ dt1 ...dtn

σ∈Sn

{ 1 E f 2 (tσ(1) , ..., tσ(n) )dt1 ...dtn ≤ n! σ∈S T n n

1 E = || f ||2L 2 (T n ) = || f ||2L 2 (T n ) . n! σ∈S n

||

8

1 Multiple Stochastic Integrals

Proposition 1.1 For m ≥ 1, the mapping Im : Em → L 2 (.) satisfies the following properties. 1. Im : Em → L 2 (.) is linear. 2. We have

Im ( f ) = Im ( f˜)

where f˜ denotes the symmetrization of f defined by (1.15). 3. We have the “isometry” of multiple integrals: for f ∈ E p , g ∈ Eq , (

{

)

E I p ( f )Iq (g) =

q!< f˜, g> ˜ L 2 (T p ) if p = q 0

(1.16)

otherwise.

In particular, for f ∈ E p , EI p ( f )2 = p!|| f˜|| L 2 (T p ) .

Proof For 1., we can always assume that f, g ∈ Em are expressed in terms of the same partition, then it is trivial from the definition (1.14) that Im (α f + g) = αIm ( f ) + Im (g). To prove 2., we can assume, by linearity, that f (t1 , ..., tm ) = 1 A1 ×....×Am (t1 , ..., tm ) with A1 , ..., Am disjoint sets in Bb . Then 1 E 1 Aσ(1) ×...×Aσ(m) (t1 , ..., tm ) f˜(t1 , ..., tm ) = m! σ∈S m

and by (1.14), 1 E Im ( f˜) = W (Aσ(1) ).....W (Aσ(m) ) m! σ∈S m

1 E W (A1 )....W ( Am ) = m! σ∈S m

= W (A1 ).....W (Am ) = Im ( f ).

1.2 Multiple Wiener-Itô Integrals

9

We now prove 3. Let f ∈ L 2 (T m ) of the form (1.13) and let g ∈ L 2 (T m ), g(t1 , .., tm ) =

N E

b j1 ,..., jm 1 A j1 ×....×A jm (t1 , ..., tm ).

j1 ,.., jm =1

Then,

=

EIn ( f )Im (g) N E

( ) ai1 ,...,in b j1 ,..., jm E W (ai1 ) . . . W ( Ain )W (A j1 ) . . . W (A jm ) .

i 1 ,..,i n , j1 ,.., jm =1

Assume m /= n, say m > n. Then there always exists k = 1, ..., m with jk ∈/ {i 1 , ..., i m } and ) ( E W (ai1 ) . . . W ( Ain )W (A j1 ) . . . W (A jm ) = E (. . .) E(W (A jk ) = 0. Let m = n. Then EIn ( f )In (g) ⎛ ⎞⎛ ⎞ E E = E ⎝n! ai1 ,..,in W (Ai1 )....W (Ain )⎠ ⎝n! b j1 ,.., jn W ( A j1 )....W (A jn )⎠ i 1 0, we decompose C1 as C1 =

N ||

E k with μ(E k ) < ε and E k disjoints for k = 1, ..., N .

k=1

Then In ( f )I1 (g) = W (A1 )....W ( Am )W (B) = (W (C1 ) + W (D1 ))W (A2 )....W (Am )(W (C1 ) + W (B0 )) = W (C1 )2 W (A2 )....W (Am ) + (W (C1 )W (B0 ) + W (C1 )W (D1 ) + W (D1 )W (B0 )) W (A2 )....W (Am ). Now, we write W (C1 ) = 2

( N E

)2 W (E k )

=

k=1

N E

W (E k )2 +

k=1

N E

W (E k )W (El ).

k,l=1;k/=l

So, Im ( f )I1 (g) =

N E k=1

W (E k )2 W ( A2 )....W (Am ) +

N E

W (E k )W (El )W (A2 )....W (Am )

k,l=1;k/=l

+ (W (C1 )W (B0 ) + W (C1 )W (D1 ) + W (D1 )W (B0 )) W (A2 )....W (Am ).

(1.20)

14

1 Multiple Stochastic Integrals

Set N E

hε =

1 Ek ×El ×A2 ×....×Am

k,l=1;k/=l

+1C1 ×B0 ×A2 ×...×Am + 1C1 ×D1 ×A2 ×...×Am + 1 D1 ×B0 ×A2 ×...×Am . By (1.20), Im ( f )I1 (g) = Im+1 (h ε ) +

N E

W (E k )2 W ( A2 )....W (Am )

k=1

= Im+1 (h ε ) +

N E (

) W (E k )2 − μ(E k ) W ( A2 )....W (Am )

k=1

+

N E

μ(E k )W (A2 )....W ( Am )

k=1

= Im+1 (h ε ) + Rε + μ(C1 )W ( A2 )....W (Am ) with Rε =

N E (

) W (E k )2 − μ(E k ) W ( A2 )....W (Am ).

k=1

Next, we observe that, by (1.18), ( f ⊗1 g)(t1 , ..., tm−1 ) { 1 E = 1 Aσ(1) ×....×Aσ(m) (t1 , ..., tm−1 , u)1 B (u)du T m! σ∈S m { 1 E 1 Aσ(1) ×....×Aσ(m−1) (t1 , ..., tm−1 ) 1 A(σ(m) (u)(1C1 (u) + 1 B0 (u))du = m! σ∈S T m

1 E 1 Aσ(1) ×....×Aσ(m−1) (t1 , ..., tm−1 )μ(C1 )1σ(m)=1 = m! σ∈S m

1 = 1 A~ ×...×Am (t1 , ..., tm−1 )μ(C 1 ). m 2 We obtained, Im ( f )I1 (g) = Im+1 (h ε ) + m Im−1 ( f ⊗1 g) + Rε .

(1.21)

1.2 Multiple Wiener-Itô Integrals

15

Let us show that Rε →ε→0 0 in L 2 (.). Since (A2 , ..., Am , E k , k ≥ 1) are disjoint, then (W ( A2 ), ..., W ( Am ), W (E k ), k ≥ 1) is a family of independent Gaussian random variables. Thus, E(Rε2 )

=E

( N E(

W (E k ) − μ(E k ) 2

)2 )

EW (A2 )2 ....EW (Am )2

k=1

= μ(A2 )....μ(Am )

N E

( )2 E W (E k )2 − μ(E k )

k=1

= 2μ(A2 )....μ( Am )

N E

μ(E k )2

k=1

≤ 2εμ( A2 )....μ(Am )

N E

μ(E k ) = 2εμ(A2 )....μ(Am )

N E

k=1

μ(C1 ).

k=1

~g in L 2 (T m+1 ). We have Let us now prove that h˜ ε converges to f ⊗ ~g||2L 2 (T m+1 ) ≤ ||h ε − h||2L 2 (T m+1 ) , ||h˜ ε − f ⊗ with h = 1C1 ×C1 ×A2 ×...×Am + 1C1 ×B0 ×A2 ×...×Am + 1C1 ×D1 ×A2 ×...×Am + 1 D1 ×B0 ×A2 ×...×Am . Therefore ~g||2L 2 (T m+1 ) ||h˜ ε − f ⊗ N E

≤ ||

k,l=1;k/=l

=

N E

1 Ek ×El ×A2 ×...×Am − 1C1 ×C1 ×A2 ×...×Am ||2L 2 (T m+1 )

μ(E k )2 μ( A2 )...μ(Am ) ≤ εμ(C1 )μ(A2 )...μ(Am ).

k=1

By taking the limit as ε → 0 in (1.21) and using the linearity of multiple stochastic integrals, we obtain, for every f ∈ Em , g ∈ E, Im ( f )I1 (g) = Im+1 ( f ⊗ g) + m Im−1 ( f ⊗1 g).

(1.22)

16

1 Multiple Stochastic Integrals

Consider now f ∈ L 2S (T m ) and g ∈ L 2 (T ) such that || f k − f || L 2 (T m ) →k→∞ 0 and ||gk − g|| L 2 (T ) →k→∞ 0. Then clearly Im ( f k )I1 (gk ) →k→∞ Im ( f )I1 (g) in L 1 (.) and E |Im+1 ( f k ⊗ gk ) − Im+1 ( f ⊗ g)|2 ˜ 2L 2 (T m+1 ) ≤ (m + 1)!|| f k ⊗g ˜ k − f ⊗g|| ˜ k − f ⊗ g||2L 2 (T m+1 ) = (m + 1)!|| f k ⊗g and since f k ⊗ gk − f ⊗ g = f k ⊗ (gk − g) + ( f k − f ) ⊗ g, we get | |2 E | Im+1 ( f k ⊗ gk ) − Im+1 ( f ⊗ g)| ≤ 2(m + 1)!|| f k ⊗ (gk − g)||2L 2 (T m+1 ) + 2(m + 1)!||( f k − f ) ⊗ g||2L 2 (T m+1 ) ≤ 2(m + 1)!|| f k ||2L 2 (T m ) ||gk − g||2L 2 (T ) + 2(m + 1)!|| f k − f ||2L 2 (T m ) ||g|| L 2 (T ) →k→∞ 0.

Similarly, by writting f k ⊗1 gk − f ⊗ g = f k ⊗1 (gk − g) + ( f k − f ) ⊗1 g, we will have | |2 E | Im−1 ( f k ⊗1 gk ) − Im−1 ( f ⊗1 g)| ≤ 2(m − 1)!|| f k ||2L 2 (T m ) ||gk − g||2L 2 (T ) + 2(m − 1)!|| f k − f ||2L 2 (T m ) ||g|| L 2 (T ) →k→∞ 0.

||

1.2.3 The Wiener Chaos The definition of the Wiener chaos is related to the Hermite polynomials. The Hermite polynomial of degree n is defined by H0 (x) = 1 for every x ∈ R and for n ≥ 1, Hn (x) =

(−1)n x 2 d n ( − x 2 ) e2 e 2 . n! dxn

The first Hermite polynomials are H1 (x) = x, H2 (x) = 21 (x 2 − 1).

(1.23)

1.2 Multiple Wiener-Itô Integrals

17 t2

Proposition 1.4 Consider the function F(x, t) = et x− 2 for x, t ∈ R. Then F(x, t) =

E

t n Hn (x).

(1.24)

n≥0

Proof It follows from the Taylor expansion around the origin of the function t → F(x, t). Indeed, for x, t ∈ R, F(x, t) = 1 + and one can show that forever n ≥ 1,

E ∂n F | t n | , ∂t n t=0 n! n≥1 |

∂n F | ∂t n t=0

= n!Hn (x).

||

We list the main properties of the Hermite polynomials: Proposition 1.5 1. For every n ≥ 1, we have Hn' (x) = Hn−1 (x)

(1.25)

2. For every n ≥ 1, we have (n + 1)Hn+1 (x) = x Hn (x) − Hn−1 (x), 3. For every n ≥ 1,

Hn (−x) = (−1)n Hn (x), n ≥ 1.

x ∈R

(1.26)

(1.27)

Proof For the first property, we differentiate ith respect to x in (1.24) and we identify the coefficients of the power series. To prove 2., we differentiate with respect to t in (1.24) and we identify the coefficients of the power series. For 3., we use F(−x, t) = || F(x, −t). Definition 1.4 Let m ≥ 0. The Wiener chaos of order m (with respect to an isonormal process W ) is defined as the vector subspace of L 2 (.) generated by the random variables (Hm (W (h)), h ∈ H, ||h|| H = 1) . By definition, since H0 (x) = 1 for every x ∈ R, the chaos of order zero is H0 = R. The chaos of order one coincides with the Gaussian space generated by (W (h), h ∈ H, ||h|| H = 1). Consequently, H1 contains only Gaussian random variables. For m ≥ 2, all the non-trivial elements of Hm are non-Gaussian random variables. The below result gives the link between the Wiener chaos and the multiple stochastic integrals.

18

1 Multiple Stochastic Integrals

Proposition 1.6 Let (W (h), h ∈ H ) be an isonormal process. Let h ∈ H with ||h|| H = 1. Then for every m ≥ 1, m!Hm (W (h)) = Im (h ⊗m ).

(1.28)

Proof The relation (1.28) will be proven by induction on m ≥ 1. For m = 1, (1.28) holds true because H1 (W (h)) = W (h) = I1 (h) by the construction of multiple integrals. Assume that (1.28) holds for 1, 2, ..m. We have, by the product formula, Im (h ⊗m )I1 (h) = Im+1 (h ⊗(m+1) ) + m Im−1 (h ⊗m ⊗1 h) = Im+1 (h ⊗(m+1) ) + m Im−1 (h ⊗(m−1) ),

(1.29)

due to the fact that h ⊗m ⊗1 h = ||h||2H h ⊗(m−1) = h ⊗(m−1) . Using the induction hypothesis, we can write by (1.29), Im+1 (h ⊗(m+1) ) = m!Hm (W (h))W (h) − m(m − 1)!Hm−1 (W (h)) = m! (Hm (W (h))W (h) − Hm−1 (W (h))) = (m + 1)!Hm+1 (W (h)) where we used (1.26) for the last identity.

||

Corollary 1 The Wiener chaos Hm coincides with the image of L 2S (T m ) through the application Im . Proof The first observation is that for every m ≥ 1, and for every h ∈ H , ( ) Hm (W (h)) ∈ Im L 2S (T m ) , EN due to Proposition 1.6, so linear combination i=1 λi Hm (W (h i )) (with λi ∈ ( any ) R, h i ∈ H ) belongs to Im L 2S (T m ) , which is a vector space. By the isometry formula ( ) (1.16) with m = n, we notice the Im L 2S (T m ) is also a closed subspace of L 2 (.). Therefore ( ) Hm ⊂ Im L 2S (T m ) . (1.30) On the by the orthogonality of multiple stochastic integrals, given ( other hand, ) F ∈ Im L 2S (T m ) , we have EF In (h ⊗n ) = 0, for n ≥ 1, n /= m.

1.2 Multiple Wiener-Itô Integrals

19

This, together with Proposition 1.6, implies that EF G = 0 for every G ∈ Hn , n ≥ 1, n /= m. ( ) So, Im L 2S (T m ) is orthogonal to any Hn , with n /= m. By (1.30), one obtains ( ) Hm = Im L 2S (T m ) . || Remark 1.5 It follows from Corollary 1 and by the isometry (1.16) that two Wiener chaoses of different order are orthogonal, i.e. if F ∈ Hm and G ∈ Hm with m /= n, then EF G = 0. The next results gives the equivalence of all norm of the Wiener chaos. It is also known as the hypercontractivity property. For the proof, see Corollary 2.8.14 in [27]. Proposition 1.7 Let F be an element in a fixed sum of Wiener chaoses, i.e. F ∈ N ⊕n=0 Hn . For for every 1 < p < q, we have ||F|| L p (.) ≤ ||F|| L q (.) ≤ C( p, q)||F|| L p (.) . In particular, for every p > 2, )p ( E|F| p ≤ C( p) E|F|2 2 .

(1.31)

Remark 1.6 For further use, let us note that ijn the particular case p = 2, the following inequality holds: for q > 2, ||F|| L q (.) ≤ (q − 1)||F|| L 2 (.) .

(1.32)

The below result is a fundamental result in the analysis on Wiener space. We refer to [29] for its proof. Theorem 1.1 (Wiener chaos expansion) Any random variable F ∈ L 2 (., G, P) (G is the sigma-algebra generated by W ) can be expanded into a series of multiple stochastic integrals E Im ( f m ), F= m≥0

with EF = I0 ( f 0 ) and f m ∈ L 2 (T m ). If the kernels f m are assumed to be symmetric, then they are uniquely determined by F.

20

1 Multiple Stochastic Integrals

1.2.4 The General Product Formula Let us now state the general product formula. Theorem 1.2 Let f ∈ L 2S (T m ) and g ∈ L 2S (T n ) with m, n ≥ 1. Then ( )( ) m n r! Im+n−2r ( f ⊗r g). Im ( f )In (g) = r r r =0 m∧n E

(1.33)

Proof Let m ≥ n. We prove formula (1.33) by induction on m ≥ 1.For m = 1, ~g2 with g1 ∈ the result has been proven in Proposition 1.3. Assume g = g1 ⊗ 2 n 2 L S (T ), g2 ∈ L (T ) with g1 ⊗1 g2 = 0. Then, by (1.3), In (g) = In−1 (g1 )I1 (g2 ) and by using the induction hypothesis, Im ( f )In (g) = Im ( f )In−1 (g1 )I1 (g2 ) ) n−1 ( )( E m n−1 r! Im+n−1−2r ( f ⊗r g1 )I1 (g2 ). = r r r =0 Now, by (1.19), Im+n−1−2r ( f ⊗r g1 )I1 (g2 ) ~r g1 )I1 (g2 ) = Im+n−1−2r ( f ⊗ ( ) ( ) ~r g1 ) ⊗ g2 + (m + n − 2r − 1)Im+n−2r −2 ( f ⊗ ~r g1 ) ⊗1 g2 = Im+n−2r ( f ⊗ Therefore Im ( f )In (g) n−1 E ( )(

) ( ) m n−1 ~r g1 ) ⊗ g2 Im+n−2r ( f ⊗ r r r=0 ) n−1 ( )( E ( ) m n−1 ~r g1 ) ⊗1 g2 r! (m + n − 2r − 1)Im+n−2r −2 ( f ⊗ + r r r=0 ( )( ) n−1 E ( ) m n−1 ~r g1 ) ⊗ g2 r! Im+n−2r ( f ⊗ = r r r=0 ( )( ) n E ( ) m n−1 ~r −1 g1 ) ⊗1 g2 + (r − 1)! (m + n − 2r + 1)Im+n−2r ( f ⊗ r −1 r −1 r=1

=

r!

We use the combinatorial identity ) ( ) r (m + n − 2r + 1) ( ~r−1 g1 ) ⊗1 g2 + (n − r ) ( f ⊗ ~r g1 ) ⊗ g2 = n( f ⊗ ~r g1 ) (f⊗ m −r −1

1.3 Random Variables in the Second Wiener Chaos

21

~g2 with g1 ∈ L 2S (T n ), g2 ∈ L 2 (T ) with g1 ⊗1 to obtain (1.33) for g = g1 ⊗ g2 = 0. || In particular, if m = n = 2 we get for f, g ∈ L 2S (T 2 ), I2 ( f )I2 (g) = I4 ( f ⊗ g) + 4I2 ( f ⊗1 g) + 2< f, g> L 2 (T 2 ) and if h, g ∈ L 2 (T ) with L 2 (T ) = 0, then Im (h ⊗m )In (g ⊗n ) = Im+n (h ⊗m ⊗ g ⊗n ).

1.3 Random Variables in the Second Wiener Chaos The random variables in the second Wiener chaos constitutes a special class among the multiple stochastic integrals. We have some knowledge about their probability distribution. Although they are non-Gaussian, we will show below that their probability law is completely determined by their cumulants (or, equivalently, by their moments). Moreover, it is possible to obtain an explicit formula for these cumulants. For f ∈ H o2 (i.e. f ∈ H ⊗2 and f is symmetric), consider the operator A f : H → H,

A f (g) = f ⊗1 g.

A classical result in functional analysis says that A f is an Hilbert-Schmidt operator. Denote by (λ j, f , j ≥ 1), (e j, f , j ≥ 1) the eigenvalues and the eigenvectors of A f . Then we have E λ j, f e j, f ⊗ e j, f . (1.34) f = j≥1

Moreover for every p ≥ 2, we have ∞ E

p

λ j, f < ∞

j=1

and p T r (A f )

= H 2 =

∞ E

p

λ j, f ,

(1.35)

j=1 p

where T r (A f ) stands for the trace of the operator A f = A f ◦ . . . A f . The sequence ~ ~~ ~ p

(f

⊗(m) 1

f, m ≥ 1) is defined recursively: f

⊗(1) 1

f = f and for p ≥ 2,

22

1 Multiple Stochastic Integrals ( p)

f ⊗1

( p−1)

f = ( f ⊗1

f ) ⊗1 f.

We deduce that the random variables have a particular form. Proposition 1.8 Let f ∈ H ⊗2 , f symmetric. Then I2 ( f ) =

E

( ) λ j, f Z 2j − 1

(1.36)

j≥1

where (Z j , j ≥ 1) is a family of independent standard normal random variables and λ j, f are defined by (1.34). Proof By (1.34), I2 ( f ) =

E

λ j, f I2 (e j, f ⊗ e j, f ) =

j≥1

E

( ) λ j, f I1 (e j, f )2 − 1 .

j≥1

It suffices to observe that (Z j = I1 (e j, f ), j ≥ 1) is a family of i.i.d. N (0, 1) random variables. || Let us denote by km (F), m ≥ 1 the mth cumulant of a random variable F. It is defined as ∂m km (F) = (−i )m m ln E(eit F )|t=0 , ∂t if F ∈ L m (.). In particular, k1 (F) = EF and k2 (F) = V ar (F). In other words, for every u ∈ R, ∞ E (iu)m (1.37) log E(eiu F ) = km (F). m! m=1 We have the following link between the moments and the cumulants of F: for every m ≥ 1, E (−1)r −1 (r − 1)!EX |a1 | . . . EX |ar | (1.38) km (F) = σ=(a1 ,..,ar )∈P({1,..,n})

if F ∈ L m (.), where P(b) is the set of all partitions of b. In particular, for centered random variables F, we have k1 (F) = EF, k2 (F) = EF 2 , k3 (F) = EF 3 , k4 (F) = EF 4 − 3(EF 2 )2 . We have an explicit expression of the cumulants of the random variables in the second Wiener chaos. Theorem 1.3 Let F = I2 ( f ) with f ∈ L 2S (T 2 ), Then for every m ≥ 2, { km (F) = 2m−1 (m − 1)!

Tm

f (u 1 , u 2 ) f (u 2 , u 3 ).... f (u m−1 , u m ) f (u m , u 1 )du 1 ...du m .

(1.39)

1.3 Random Variables in the Second Wiener Chaos

23

Proof The proof is taken from [27]. First, from (1.36), we can compute the characteristic function of F, obtaining,for u ∈ R, E(eiu F ) =

∞ ||

/

j=1

e−iuλ j, f . 1 − 2iuλ j, f

By applying the logarithm above, we find log E(eiu F ) =

∞ E m=2

2m−1

∞ (iu)m E p λ . m j=1 j, f

(1.40)

By identifying the coefficients in (1.37) and (1.40), we get k1 (F) = 0 and for m ≥ 2, km (F) = 2m−1 (m − 1)!

E

λmj, f = 2m−1 (m − 1)!< f ⊗1(m−1) f, f > H ⊗2

j≥1

where H = L 2 (T ) and the last equality comes from (1.35). It then remains to check that for m ≥ 2, (m−1)

< f ⊗1

{ f, f > H ⊗2 =

Tm

f (x1 , x2 ) f (x2 , x3 ) . . . f (xm−1 , xm ) f (xm , x1 )d x1 . . . d xm . (1.41)

To this end, we prove first by induction that for m ≥ 2, ( f ⊗(m) f )(x1 , x2 ) = 1

{ T m−1

du 1 ...du m−1 f (x1 , u 1 ) f (u 1 , u 2 ).... f (u m−2 , u m−1 ) f (u m−1 , x2 ).

(1.42)

Formula (1.42) holds for m = 2, due to (1.18). Then ( ) f ) ⊗ f (x1 , x2 ) ( f ⊗1(m+1) f )(x1 , x2 ) = ( f ⊗(m) 1 1 { = du m ( f ⊗(m) f )(x1 , u m ) f (x2 , u m ) 1 T { { du m f (x2 , u m ) du 1 ...du m−1 f (x1 , u 1 ) f (u 1 , u 2 ).... f (u m−1 , u m ) = T m−1 {T du 1 ...du m f (x1 ), u 1 ) f (u 2 , u 3 ) . . . f (u m−1 , u m ) f (u m , x2 ). = Tm

This implies (1.42). It then suffices to use (1.42) to obtain (1.41) and the conclusion. || Finally, let us show that the cumulants (or the moments, see relation (1.38)) determines the law of a random variable in the second Wiener chaos. We say that the U the law of a random variable Y ∈ m≥1 L m (.) is determined by the moments (or by

24

1 Multiple Stochastic Integrals

U the cumulants) if for any other random variable X ∈ m≥1 L m (.) with EY m = EX m for every m ≥ 1, we have that the law of Y coincides with the law of X . Proposition 1.9 Assume f ∈ H o2 and let F = I2 ( f ). Then the law of F is entirely determined by the sequence of cumulants (1.39). Proof We will give the sketch of the proof. Actually it suffices to show that (see Exercise 2.7.12 in [27]) Eet|X | < ∞ for some t > 0. By assuming EF 2 = 1, we have by hypercontractivity (inequality (1.32)), (

E|F| p

) 1p

≤ p−1

for every p > 2. So, by using Markov’s inequality, P (|F| > u) ≤ u − p ( p − 1) p for any u > 0. We write now p = 1 + { E(e

t|F|

)=1+t

u e

∞

with 0 < e < u. Then, via Fubini, etu P (|F| > u) du < ∞

0

for 0 ≤ t < 1e .

||

Chapter 2

Hermite Processes: Definition and Basic Properties

Historically, the Hermite processes appeared as limit of partial sums of sequences of correlated random variables in the so-called Non-Central Limit Theorem, see e.g. [17, 41, 43]. Let us briefly recall the basic facts. Let Z ∼ N (0, 1) and consider a function g : R → R such that Eg(Z ) = 0 and Eg(Z )2 < ∞. Then g can be expanded into a basis of Hermite polynomial, i.e. g(x) =

∞ ∑

c j H j (x),

x ∈ R,

j=0

where H j denotes the jth Hermite polynomial given by (1.23) and c j = E(g(Z )H j (Z )). The Hermite rank of g is defined by | k = min{l |cl /= 0}. Since E [g(Z )] = 0, we have k ≥ 1. Assume g has Hermite rank equal to q and let (ξn , n ∈ Z) be a stationary Gaussian sequence with mean 0 and variance 1 which exhibits long range dependence in the sense that the correlation function satisfies r (n) := E (ξ0 ξn ) = n

2H −2 q

L(n)

where H ∈ ( 21 , 1), q ≥ 1 and L is a slowly varying function at infinity (see e.g. [18] for the definition). Consider the partial sums, for n ≥ 1, t ≥ 0, X n (t) =

[nt] 1 ∑ g(ξ j ). n H j=1

(2.1)

Due to the relatively strong correlation of the summands of the above sum, the sequence (X n , n ≥ 1) will have, in many situations, a non-Gaussian limit. Actually, the family of stochastic processes (X n , n ≥ 1) converges as n → ∞, in the sense © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Tudor, Non-Gaussian Selfsimilar Stochastic Processes, SpringerBriefs in Probability and Mathematical Statistics, https://doi.org/10.1007/978-3-031-33772-7_2

25

26

2 Hermite Processes: Definition and Basic Properties

of finite-dimensional distributions, to a stochastic process which lives in the Wiener chaos of order q. This limit process is the Hermite process. Below, we will define it properly and we will analyze its main properties. The class of Hermite processes contains the well-known fractional Brownin motion, which is the only Gaussian Hermite process. We will see that the Hermite process (of general order) shares many properties with fBm: covariance structure, self-similarity, stationarity of the increments, regularity of sample paths. Therefore, the Hermite process constitutes an interesting alternative to fBm for modelling purposes, especially in applications where the empirical data shows a non-Gaussian character. An example of such an application in hidrology has been provided in [42]. More recently, the Hermite processes found applications in network traffic (see e.g. [9]) or mathematical finance (see e.g. [19, 40]). Notice that a generalized version of the Hermite process (whose self-similarity index may be less than one half) has been studied in e.g. [7] or [3]. the contain of this chapter. In the sequel we will denote by ) ( Let us describe H,q Z t , t ≥ 0 the Hermite process of order q ≥ 1 with Hurst parameter (or self( ) similarity index) H ∈ 21 , 1 . • We start with an analysis of the kernel of the Hermite process. Being an eleH,q ment of the qth Wiener chaos, the random variable Z t can be expressed as a H,q H,q multiple integral of order q, i.e. Z t = Iq (L t ) for every t ≥ 0, according to H,q Corollary 1. The function L t is defined on Rq and it is called the kernel of the Hermite process. The properties of this kernel play an important role for the stochastic analysis of the Hermite process. • We then study the main properties of the Hermite process such as the scaling property, the stationary of its increments, the moments, the regularity of the sample paths or the p-variation. • We included a paragraph consecrated to some particular Hermite processes: the fractional Brownian motion (which is the Hermite process of order 1 and the only Gaussian Hermite process) and the Rosenblatt process (the Hermite process of order q). For these particular case, it is possible to completely describe their finite-dimensional probability distributions by using their cumulants. • Other integral representations of the Hermite process are also stated and proven. H,q H,q Actually, the multiple integral representation Z t ) = Iq (L t ) is not unique. We ( H,q

show that the stochastic process Z t , t ∈ [0, T ] coincides, in the sense of finite) ( H,q H,q dimensional distributions, with Iq (At ), t ∈ [0, T ] , where At is a kernel in H,q

L 2 ([0, T ]q ) different from L t . Depending of the problem considered, one or another representation may be more useful. • A short discussion on the possibility to simulate the Hermite processes is included in the last part of this chapter.

2.1 The Kernel of the Hermite Process

27

2.1 The Kernel of the Hermite Process We start by introducing the kernel of the Hermite process. For a, b > 0, we denote by β(a, b) the beta function (

1

β(a, b) =

x a−1 (1 − x)b−1 d x.

(2.2)

0

Let us start with the following lemmas which will be intensively used in the sequel. ( ) Lemma 2.1 Let H ∈ 21 , 1 . Then for every s, t ≥ 0, H (2H − 1)

( t( 0

s

|u − v|2H −2 dudv =

0

) 1 ( 2H t + s 2H − |t − s|2H . 2

Proof The formula can be obtained by differentiation both sides

∂2 . ∂s∂t

∎

Lemma 2.2 Let a, b, u, v ∈ R such that a + b + 1 < 0 and u /= v. Then (

u∧v

−∞

{

β(−1 − a − b, a + 1)(v − u)a+b+1 , if u < v β(−1 − a − b, b + 1)(u − v)a+b+1 , if v < u.

(u − y)a (v − y)b dy =

In particular, if a = b < − 21 and u /= v, (

u∧v −∞

(u − y)a (v − y)a dy = β(−1 − 2a, a + 1)|u − v|2a+1 .

(2.3)

Proof We recall the alternative definition of the beta function: if x, y > 0, then ( β(x, y) =

∞

t x−1 (1 + t)−x−y dt.

0

Let u < v. By the change of variables w = (

u

we get (

(u − y) (v − b) dy = (v − u) a

−∞

u−y , v−u

b

a+b+1

∞

wa (1 + w)b

0

= β(a + 1, −1 − a − b)(v − u)a+b+1 . ∎ For t ≥ 0, we set ( H,q L t (y1 , . . . , yq )

t

= c(H, q) 0

(u −

) ( − 21 + 1−H q y1 )+

. . . (u −

) ( − 21 + 1−H q yq )+ du,

(2.4)

28

2 Hermite Processes: Definition and Basic Properties

for every y1 , . . . , yq ∈ R, with c(H, q)2 =

( q!β

H,q

Proposition 2.1 For t ≥ 0, let L t every s, t ≥ 0, H,q

⟨L t

H (2H − 1) 1 2

−

1−H 2−2H , q q

)q .

(2.5)

H,q

be given by (2.4). Then L t

, L sH,q ⟩ L 2 (Rq ) = q!

∈ L 2 (Rq ) and for

) 1 ( 2H t + s 2H − |t − s|2H . 2

In particular, for every t ≥ 0, H,q 2 || L 2 (Rq )

||L t

= q!t 2H .

Proof We have, for s, t ≥ 0, by Fubini, ( H,q ⟨L t ,

L sH,q ⟩ L 2 (Rq )

=

((

( = c(H, q) ((

s

×

2 Rq

(v −

0

( = c(H, q)

2

H,q

Lt

Rq

t

dy1 ...dyq

(y1 , ..., yq )L sH,q (y1 , ..., yq )dy1 . . . dyq

(u −

0 ) ( − 21 + 1−H q y1 )+

(

t

du

dv

0

R

0

We apply (2.3) with a = − 21 + ( (u −

( =

. . . (v −

((

s

R u∧v −∞

(

=β

) ( − 21 + 1−H q y1 )+

(u −

H −1 q

) ) ( − 21 + 1−H q yq )+ dv

) ( − 21 + 1−H q y)+ (v

−

)q ) ( − 21 + 1−H q y)+ dy

< − 21 to get

) ( − 21 + 1−H q y)+ (v

(u −

. . . (u −

) ) ( − 21 + 1−H q yq )+ du

−

( ) − 21 + 1−H q (v y)+

1 1 − H 2 − 2H − , 2 q q

) ( − 21 + 1−H q y)+ dy

− )

( ) − 21 + 1−H q y)+ dy

|u − v|

2H −2 q

.

.

2.2 Definition of the Hermite Process and Some Immediate Properties

29

Thus ) ( ( ( s 1 1 − H 2 − 2H q t ⟩ L 2 (Rq ) = c(H, q)2 β − , du dv|u − v|2H −2 q q 2 0 0 ) ( ) 1 1 ( 2H 1 1 − H 2H − 2 q t − , + s 2H − |t − s|2H = c(H, q)2 β q q H (2H − 1) 2 2 ) 1 1 ( 2H = t + s 2H − |t − s|2H , q! 2 H,q

⟨L t

H,q

, Ls

∎

where we used (2.5). Also notice that the kernel (2.4) satisfies the following scaling property q

L ct (y1 , ..., yq ) = c H − 2 L t H,q

H,q

(y

1

c

, ...,

yq ) , c

for every c > 0 and for every y1 , ..., yq ∈ R. This property plays a role in the proof of the self-similarity of the Hermite process.

2.2 Definition of the Hermite Process and Some Immediate Properties We start with the definition of the Hermite process. Definition 2.1 The Hermite process ( of order )q ≥ 1 and with self-similarity index ( ) H,q H ∈ 21 , 1 is defined as Z H,q = Z t , t ≥ 0 , with H,q

Zt

((

(

= c(H, q) H,q

= Iq (L t

Rq

),

t

(u −

0

( ) − 21 + 1−H q y1 )+

. . . (u −

) ( ) − 21 + 1−H q yq )+ du d B(y1 ) . . . d B(yq )

t ≥ 0,

(2.6)

where B = (B(y), y ∈ R) is a Brownian motion over the whole real line (or the twosided Wiener process, see (1.5)), Iq is the multiple integral of order q with respect to B defined in Sect. 1.2 and the kernel L H,q is defined by (2.4). H,q

Notice that for every t ≥ 0, the random variable Z t to L 2 (Ω), since (see Proposition 2.1) H,q 2

E(Z t

H,q 2 || L 2 (Rq )

) = q!||L t

is well-defined and belongs

= t 2H .

30

2 Hermite Processes: Definition and Basic Properties

Moreover, for every s, t ≥ 0, the covariance of the process Z H,q reads as, H,q

EZ t

Z sH,q = R H (t, s) :=

) 1 ( 2H t + s 2H − |t − s|2H . 2

(2.7)

A Hermite random variable will be in the sequel a random variable (with index H H,q and order q) with the same distribution as Z 1 .

2.2.1 Self-similarity and Stationarity of the Increments We will say that a stochastic process (X t , t ≥ 0) is self-similar of index H ∈ (0, 1) (or H -self-similar) if for every c > 0, the stochastic processes (X ct , t ≥ 0) and (c H X t , t ≥ 0) have the same finite-dimensional distributions. A stochastic process (X t , t ≥ 0) is said to be with stationary increments if for every h > 0, the stochastic processes (X t , t ≥ 0) and (X t+h − X h , t ≥ 0) have the same finite-dimensional distributions. We will check these properties for the Hermite process. Proposition 2.2 The process Z H,q given by (2.6) is self-similar of index H and it has stationary increments. Proof Let c > 0. Then for every t ≥ 0, H,q Zt = c(H, q)

( Rq

(( 0

ct

(

− (u − y1 )+

1 1−H 2+ q

)

(

− . . . (u − yq )+

1 1−H 2+ q

)

)

du

×d B(y1 ) . . . d B(yq ) ) ) ) ( ( ( (( t − 21 + 1−H − 21 + 1−H q q . . . (cu − yq )+ du (cu − y1 )+ = c × c(H, q) Rq

0

×d B(y1 ) . . . d B(yq ) ) ) ) ( ( ( (( t − 21 + 1−H − 21 + 1−H q q . . . (cu − cyq )+ du (cu − cy1 )+ = c × c(H, q) Rq

0

×d B(cy1 ) . . . d B(cyq ) ( ((

q = c− 2 +H c(H, q)

Rq

×d B(cy1 ) . . . d B(cyq ).

0

t

(

− (u − y1 )+

1 1−H 2+ q

)

(

− . . . (u − yq )+

1 1−H 2+ q

)

)

du

2.2 Definition of the Hermite Process and Some Immediate Properties

31

Since the Brownian motion B is 21 -self-similar, we have (we denote by ≡(d) the equivalence of finite-dimensional distributions) ((

(( Rq

t

(cu −

0

( ) − 21 + 1−H q cy1 )+

. . . (u −

)) ( ) − 21 + 1−H q yq )+ du

) ×d B(cy1 ) . . . d B(cyq ), t ≥ 0 )) (( (( ( ( ) ) t − 21 + 1−H − 21 + 1−H q q (d) q2 . . . (u − yq )+ du (u − y1 )+ ≡ c Rq

0

) d B(y1 ) . . . d B(yq ), t ≥ 0 . Thus, for every c > 0, (

) ) ( H,q H,q Z ct , t ≥ 0 ≡(d) c H Z t , t ≥ 0 .

To prove the stationarity of the increments, we write, for h > 0 H,q

H,q

Z t+h − Z t ( (( = c(H, q) Rq

t+h

(u −

t

( ) − 21 + 1−H q y1 )+

. . . (u −

) ( ) − 21 + 1−H q yq )+ du

×d B(y1 ) . . . d B(yq ) ) ) ) ( ( ( (( t − 21 + 1−H − 21 + 1−H q q (u − (y1 − h))+ . . . (u − (yq − h))+ du = c(H, q) Rq

0

×d B(y1 ) . . . d B(yq ) ) ( ( ) ) ( (( t − 21 + 1−H − 21 + 1−H q q . . . (u − yq )+ du (u − y1 )+ = c(H, q) Rq

≡(d)

0

×d B(y1 + h) . . . d B(yq + h) ) ) ) ( ( ( (( t − 21 + 1−H − 21 + 1−H q q c(H, q) (u − y1 )+ . . . (u − yq )+ du Rq

0

×d B(y1 ) . . . d B(yq ). The last relation follows from the definition of the multiple stochastic integrals and from the fact that (B(y + h) − B(h), y ∈ R) has the same finite dimensional distributions as (B(y), y ∈ R). ∎

32

2 Hermite Processes: Definition and Basic Properties

2.2.2 Moments and Hölder Continuity The scaling property of the Hermite process implies that for every t ≥ 0, and for every p ≥ 1, H,q H,q E|Z t | p = t H p E|Z 1 | p and by using the stationarity of the increments, for every 0 ≤ s ≤ t, and p ≥ 1, |p | | | | | H,q | p | H,q H,q E |Z t − Z sH,q | = E |Z t−s | = |t − s| H p E|Z 1 | p .

(2.8)

Via the Kolmogorov continuity criterion, we get the existence of a modification of Z H,q , which has Hölder continuous paths of order δ, for any δ ∈ (0, H ).

2.2.3 The Hermite Noise and the Long Memory Consider the sequence (X j , j ≥ 0) given by H,q

H,q

X j = Z j+1 − Z j

,

j ∈ N.

(2.9)

Then, for any j, k ∈ N, ( )( ) H,q H,q H,q H,q EX j X k = E Z j+1 − Z j Z k+1 − Z k =

) 1( | j − k + 1|2H + | j − k − 1|2H − 2| j − k|2H =: r ( j − k), 2

where r( j) =

) 1( | j + 1|2H + | j − 1|2H − 2| j|2H , 2

j ∈ Z.

For | j| large, r ( j ) behaves as H (2H − 1)| j|2H −2 . In particular, since H ∈ ∑

|r ( j )| diverges.

(1 2

) ,1 ,

(2.10)

j∈Z

The property (2.10) is usually interpreted by saying that the Hermite process Z H,q has long memory (or long range dependence).

2.2 Definition of the Hermite Process and Some Immediate Properties

2.2.4

33

p-Variation

For t > 0, let VN(2) (Z H,q )t =

N −1 ( ∑

H,q

H,q

Z ti+1 − Z ti

)2

i=0

with ti =

it N

for i = 0, ..., N . By (2.8), H,q

EVN (Z H,q )t = E|Z 1 |2

N −1 ∑

|ti+1 − ti |2H = t 2H N 1−2H → N →∞ 0.

i=0

) ( This implies that for every t > 0, the sequence VN(2) (Z H,q )t , N ≥ 1 converges to 0 in L 1 (Ω), i.e. the Hermite process is a zero quadratic variation process. In particular, this also implies that the Hermite process is not a semimartingale. We can also state a result concerning the p -variation of the Hermite process. For p ≥ 1, let N −1 | |p ∑ | H,q ( p) H,q | (2.11) VN (Z H,q )t = | Z ti+1 − Z ti | i=0

with ti , i = 0, .., N as before. The behavior of the sequence (2.11) is based on the fact that the Hermite noise defined by (2.9) is ergodic, and in particular it satisfies (see [37]) N 1 ∑ f (X j ) → N →∞ E f (X 1 ) almost surely and in L 1 (Ω) N j=1

(2.12)

for every measurable function f : R → R such that E| f (X 1 )| < ∞. ( p)

Proposition 2.3 Let t > 0. Consider the sequence (VN (Z H,q )t , N ≥ 1 given by (2.11). Then in probability

( p)

VN (Z H,q )t → N →∞

⎧ 1 ⎪ ⎨0, if p > H H,q 1 tE|Z 1 | H if p = ⎪ ⎩ +∞ if p < H1 .

1 H

Proof We define the sequence N,p

Yt

( p)

= N p H −1 VN (Z H,q )t .

By Proposition 2.2, we have (=(d) means equality in distribution in the sequel),

34

2 Hermite Processes: Definition and Basic Properties N,p

Yt

=(d) t H p

N −1 1 ∑ H,q H,q |Z − Zi | p. N j=0 i+1

The ergodicity of the sequence (2.9) implies that (see (2.12)) N −1 1 ∑ H,q H,q H,q |Z − Z i | p → N →∞ E|Z 1 | p almost surely and in L 1 (Ω). N j=0 i+1 N,p

Consequently, (Yt , N ≥ 1) converges in law (and thus in probability) as N → ∞ H,q ∎ to t H p E|Z 1 | p . This fact easily gives the conclusion.

2.2.5 Approximation by Semimartingales For ε > 0, define H,q,ε

Zt

= c(H, q)

((

( Rq

t

(u −

0

) ( − 21 + 1−H q y1 )+

d B(y1 ) . . . d B(yq ),

. . . (u −

) ) ( − 21 + 1−H q yq )+ du1u−y1 >ε ....1u−yq >ε

t ≥ 0.

(2.13)

We will show that (Z H,q,ε , ε > 0) constitutes a family of semi-martingales (more exactly, of finite-variation processes) and that for every t ≥ 0, Z H,q,ε converges to H,q Z t in L 2 (Ω). We first notice that by using Fubini theorem, ( H,q,ε

Zt

t

=

X ε,u du 0

with

( X ε,u =

q ⊓ Rq j=1

(u −

) ( − 21 + 1−H q y j )+ 1u−y j >ε d B(y1 ) . . . d B(yq ).

Indeed, we can write X ε,u = Iq (gε,u ) with gε,u (y1 , ..., yq ) = c(H, q)

q ⊓

(u −

) ( − 21 + 1−H q 1u−y j >ε , y j )+

j=1

and via the isometry property (1.16), we have for 0 ≤ u ≤ t

(2.14)

2.2 Definition of the Hermite Process and Some Immediate Properties 2 EX ε,u = q!||gε,u ||2L 2 (Rq ) (( ( = q!c(H, q)2 dy1 ...dyq Rq

= q!c(H, q)2

u−ε

−∞

(u −

−2 y)+

35

(

1−H 1 2+ q

)q

)

dy

2(1−H ) q ε− q < ∞. 2(1 − H )

H,q,ε

, t ≥ 0) is a semiConsequently, by (2.14), for every ε > 0, the process (Z t martingale. We will show that it approximates the Hermite process Z H,q at any fixed time t. Proposition 2.4 For every t ≥ 0, H,q,ε

Zt

H,q

→ε→0 Z t

in L 2 (Ω).

Proof We have | |2 | | | H,q,ε | H,q,ε |2 H,q | H,q,ε H,q E |Z t − Z t | = E |Z t Z t + t 2H . | − 2EZ t By proceeding as in the proof of Proposition 2.1, we find | | | H,q,ε |2 E |Zt | = q!c(H, q)2

( t(

((

t

(u−ε)∧(v−ε)

dudv 0

−∞

0

(u − y)

) ( − 21 + 1−H q

(v − y)

) ( − 21 + 1−H q

)q dy

and H,q,ε

EZ t

H,q

Zt

= q!c(H, q)2

( t(

((

t

(u−ε)∧v

dudv 0

−∞

0

(u − y)

( ) − 21 + 1−H q

(v − y)

( ) − 21 + 1−H q

)q dy

.

By the dominated convergence theorem and relation (2.3), both quantities above converge, as ε → 0, to ( t( 2

((

t

u∧v

dudv

q!c(H, q)

0

0

−∞

(u − y)

( ) − 21 + 1−H q

(v − y)

( ) − 21 + 1−H q

)q dy

= t 2H . ∎

36

2 Hermite Processes: Definition and Basic Properties

2.3 Some Particular Hermite Processes: Fractional Brownian Motion and the Rosenblatt Process The first two Hermite processes are the fractional Brownian motion and the Rosenblatt process. We focus on them in this paragraph.

2.3.1 Fractional Brownian Motion The fractional Brownian motion is obtained by taking q = 1 in (2.6). That is, for ) ( H ∈ 21 , 1 , the fBm (BtH , t ≥ 0) with Hurst parameter H is given by ( (( BtH = c(H, 1)

R

t

0

H − 23

(u − y)+

) du d B(y),

t ≥ 0.

with c(H, 1) from (2.5). Since the above integral is a standard Wiener integral with respect to the Wiener process (see Sect. 1.1.1, it follows that B H is a centered Gaussian process with covariance (2.7). It is well-known that the fBm can actually be defined for all H ∈ (0, 1). In this case, the covariance function (2.7) characterizes the law of the process B H , which is not true when q ≥ 2. An equivalent definition of the fBm is as the only self-similar Gaussian process with stationary increments. For H ∈ (0, 1), H /= 21 , the fBm is neither a semimartingale, nor a Markov process.

2.3.2 The Rosenblatt Process The Rosenblatt process is obtained by taking q = 2 in the relation (2.6), so ( ( (( Z tH,2

:= Z t = c(H, 2)

R

R

t 0

(s −

− 2−H y1 )+ 2

(s −

− 2−H y2 )+ 2

) ds d B(y1 )d B(y2 )

(2.15) where (B(y), y ∈ R) is a standard Brownian motion on R and c(H, 2) is defined in (2.5). The Rosenblatt process lives in the second Wiener chaos. A particularity of the Rosenblatt process is that its probability law is determined by its cumulants. We recall that when G = I2 ( f ) is a multiple integral of order 2 with respect to a Wiener sheet (B(y), y ∈ R), then its cumulants can be computed via the formula (1.39) in Theorem 1.3. Let λ1 , ..., λ N ∈ R and t1 , ..., t N ≥ 0. Let V = λ1 Z tH,2 + . . . λ N Z tH,2 . 1 N

2.3 Some Particular Hermite Processes: Fractional Brownian Motion …

37

Then the cumulants of the random variable V , which characterize the finite dimensional distributions of the Rosenblatt process, can be computed as follows. k1 (V ) = EV = 0,

k2 (V ) = V ar (V ) =

N ∑

( ) H,2 λi λ j E Z tH,2 Z tj i

i, j=1

=

N ∑

λi λ j R H (ti , t j ),

i, j=1

with R H given by (2.7). For m ≥ 3, since ⎛ ⎞ N ∑ ⎠, λ j L tH,2 V = I2 ⎝ j j=1

we find, by using (1.39), km (V )

( = 2m−1 (m − 1)! dy1 . . . dym Rm ⎛ ⎞⎛ ⎞ ⎛ ⎞ N N N ∑ ∑ ∑ ×⎝ L tH,2 (y1 , y2 )⎠ ⎝ L tH,2 (y2 , y3 )⎠ . . . ⎝ L tH,2 (ym , y1 )⎠ j j j 1

j1 =1

j2 =1

2

jm =1

( = 2m−1 (m − 1)!c(H, 2)m ×

(( t j1

H

(u 1 − y1 )+2

0

(( t j2 0

×...

H

(u 2 − y2 )+2

(( t jm 0

−1

Rm

H

−1 H

(u 1 − y3 )+2

H

(u m − ym )+2

−1

λ j1 ...λ jm

j1 ,..., jm =1

(u 1 − y2 )+2

−1

N ∑

dy1 . . . dym

m

)

du 1

−1

) du 2

H

(u m − y1 )+2

−1

) du m .

By Fubini, N ∑

km (V ) = 2m−1 (m − 1)!c(H, 2)m ×

j1 ,..., jm =1

m (( ⊓ a=1

λ j1 ...λ jm

( tj 1

H

R

(u a − y)+2

−1

H

(u a+1 − y)+2

−1

0

) dy

du 1 . . .

( tj m 0

du m

38

2 Hermite Processes: Definition and Basic Properties

with the convention u m+1 = u 1 . Lemma 2.2 gives km (V )

(

= 2m−1 (m − 1)!c(H, 2)m β ×

( tj 1 0

du 1 ...

( tj m 0

)m H ,1 − H 2

N ∑

λ j1 ...λ jm

j1 ,..., jm =1

du m |u 1 − u 2 | H −1 |u 2 − u 3 | H −1 ...|u m−1 − u m | H −1 |u m − u 1 | H −1 N ∑

m

m

= 2 2 −1 (m − 1)!(H (2H − 1)) 2

( tj 1

λ j1 ...λ jm

0

j1 ,..., jm =1

du 1 . . .

( tj m 0

×|u 1 − u 2 | H −1 |u 2 − u 3 | H −1 . . . |u m−1 − u m | H −1 |u m − u 1 | H −1 .

du m

(2.16)

2.4 Alternative Representation Here, the purpose is to give a finite-interval representation of the Hermite process, i.e. to express it as a multiple Wiener-Itô integral with respect to a Wiener process index by an interval [0, T ] with T > 0. Actually, the construction of the stochastic calculus with respect to the fractional Brownian motion is based on such a finite-time interval representation (see e.g. [29]). Let us consider the function f H (t, s) =

( s ) 21 −H t

H − 23

(t − s)+

,

s, t ≥ 0,

( ) H,q and define the process Yt , t ≥ 0 given by ((

( H,q Yt

= c(H, q)

t

f [0,t]q

H'

(u, y1 ) . . . f

H'

) (u, yq )du dW (y1 ) . . . dW (yq )

0

(2.17) where W = (W (y), y ∈ R+ ) is a (standard) Wiener process, c(H, q) is from (2.5) and H −1 . (2.18) H' = 1 + q We can also write

H,q

Yt

H,q

= Iq (lt

)

2.4 Alternative Representation

39

with H,q

lt

(y1 , ..., yq )

(

t

(y1 , ..., yq )

'

'

f H (u, y1 ) . . . f H (u, yq )du 0 ⎛ ⎞ ( t ⊓ q ( ) 21 −H ' ' 3 y H − j ⎝ = c(H, q)1[0,t]q (y1 , ..., yq ) (u − y j )+ 2 ⎠ u 0 j=1 = c(H, q)1

[0,t]q

= c(H, q)1[0,t]q (y1 , ..., yq )(y1 ....yq )− 2 + 1

1−H q

(

t

q

du u 2 −1+H

q ⊓

0

(u − y j )− 2 + 1

H −1 q

.

j=1

) ( H,q,ε , t ≥ 0 given by (2.13). By (2.14), we can write Consider the process Z t

H,q,ε Zt = c(H, q)

= c(H, q)

( t

⎛

(

du ⎝

Rq j=1

0

( t 0

q ⊓

(

(

− (u − y j )+

1 1−H 2+ q

⎞

)

)

1u−y j >ε d B(y1 ) . . . d B(yq )⎠

( ) du Iq ( f uH,ε )⊗q

where

) ( − 21 + 1−H q

f uH,ε (y) = (u − y)+

1u−y>ε .

Notice that for every ε > 0, f uH,ε ∈ L 2 (R), see Sect. 2.2.5. Let us also introduce the regularized fractional process B˙ uH,ε

( =

R

(u −

( ) − 21 + 1−H q y)+ 1u−y>ε d B(y).

(2.19)

Since for every h ∈ L 2 (R) with ||h|| L 2 (R = 1, we have by Proposition 1.6 Iq (h ⊗q ) = q!Hq (I1 (h))

(2.20)

where Hq is the qth Hermite polynomial, we get ⎛

( H,q,ε

Zt

t

= q!c(H, q) 0

) ( ⎜ du E( B˙ uH,ε )2 Hq ⎝ ( q 2

⎞ B˙ uH,ε E( B˙ uH,ε )2

⎟ ) 21 ⎠ .

(2.21)

( ) Lemma 2.3 For every T > 0, the process B˙ uH,ε , u ∈ [0, T ] given by (2.19) has ) ( H,ε the same finite dimensional distributions as the process X u , u ∈ [0, T ] , where

40

2 Hermite Processes: Definition and Basic Properties

( X uH,ε =

∞ 0

'

'

1 dW (z). z 2 −H (1 − uz) H − 2 dz1z< ε+s 1

3

Proof Notice that both B˙ H,ε and X H,ε are centered Gaussian processes, thus it suffices to check that their covariances coincide. For 0 ≤ v < u ≤ T , we have E B˙ uH,ε B˙ vH,ε

( =

(u −

R ( v−ε

=

) ( − 21 + 1−H q y)+ 1u−y>ε (v ) ( − 21 + 1−H q

(u − y) −∞ ( ' = (u − v)2H −2

∞

0

'

(v − y)

( 0

) ( − 21 + 1−H q

'

dy

ε dz z H − 2 (1 + z) H − 2 1z> u−v 3

where we performed the change of variables z = ables z = 1−x , we get x ' E B˙ uH,ε B˙ vH,ε = (u − v)2H −2

−

) ( − 21 + 1−H q y)+ 1v−y>ε dy

∞

v−y . u−v

3

With the new change of vari-

'

'

u−v d x. x 1−2H (1 − x) H − 2 1x< ε−(u−v)

Thew two successive changes of variable x =

a(u−v) v(u−a)

3

and a = uvb give

( v ' ' 3 ' 3 H,ε ˙ H,ε H ' − 21 ˙ E Bu Bv = (uv) a 1−2H (v − a) H − 2 (u − a) H − 2 1a< u−v ε+v 0 ( ∞ ' ' 3 ' 3 1 1 1 db = b1−2H (1 − bu) H − 2 (1 − bv) H − 2 1b< ε+u b< ε+v 0

= EX uH,ε X vH,ε .

∎

By (2.21) and Lemma 2.3, we can write H,q,ε

Zt

≡(d) q!c(H, q)

⎞

⎛

(

t 0

) ( ⎟ ⎜ du E(X uH,ε )2 Hq ⎝ ( ) 21 ⎠ . E(X uH,ε )2 q 2

X uH,ε

Via (2.20), for T > 0, (

H,q,ε

Zt

) ) ( H,q,ε , t ∈ [0, T ] ≡(d) X t , t ∈ [0, T ]

(2.22)

2.4 Alternative Representation

41

with = c(H, q)

⎛

⎛

(

( H,q,ε Xt

t

⎝ (0,∞)q

du ⎝

q ⊓

0

⎞⎞ 1 ' 2 −H

zj

(1 − uz j ) H

'

− 23

1 ⎠⎠ dz1z j < ε+s

j=1

×dW (z 1 ) . . . dW (z q ). Let us state and prove a first representation of the Hermite process, different by (2.6). ( ) H,q Proposition 2.5 For every T > 0, the Hermite process Z t , t ∈ [0, T ] has the ) ( H,q same finite dimensional distributions as the process X t , t ∈ [0, T ] given by H,q Xt = c(H, q)

⎛

⎛

(

( t

(0,∞)q

du ⎝

⎝

0

q ⊓

⎞⎞ 1 ' 2 −H

zj

H '− 3 (1 − uz j )+ 2 dz ⎠⎠ d W (z 1 ) . . . d W (z q ).

j=1

Proof Let us first notice that the process X H,q is well-defined. Indeed, for every t ≥ 0, H,q 2

E(X t

)

⎛

( = q!c(H, q)2

(

t

⎝

du

(0,∞)q

0

= q!c(H, q)2 lim ) ε→0

(

⎝ (0,∞)q

ε→0 Rq

⎛

= q!c(H, q)2

⎝

⎝ Rq

0 j=1

H

(1 − uz j )+

q ⊓

'

− 23

⎠ dz 1 ...dz j ⎞2

1 ' 2 −H

zj

(1 − uz j ) H

j=1

'

− 23

1 ⎠ dz 1 ...dz j 1z j < ε+u

⎞2

(u − x j ) H

0 j=1

( t⊓ q

(

du 0

( t⊓ q

= q!c(H, q)2 lim

zj

t

⎛

(

⎞2 1 ' 2 −H

j=1

⎛

(

q ⊓

'

− 23

1u−x j >ε ⎠ d x1 ...d xq

⎞2 H0 − 23

(u − x j )+

⎠ d x1 ...d xq .

We know from Proposition 2.4 that for every t ≥ 0, H,q,ε

Zt

H,q

→ε→0 Z t

in L 2 (Ω).

Using the same arguments, it is easy to show that for every t ≥ 0, H,q,ε

Xt

H,q

→ε→0 X t

in L 2 (Ω).

We conclude by taking the limit as ε → 0 in (2.22).

∎

42

2 Hermite Processes: Definition and Basic Properties

Now, set H,q,ε

Wt

= c(H, q)

⎛

⎛

(

(

t

⎝ (0,∞)q

du ⎝

0

q ⊓

⎞⎞ 1 ' 2 −H

zj

(1 − uz j ) H

'

− 23

⎠⎠ dz1z j < 1−ε s

j=1

×dW (z 1 ) . . . dW (z q ) and observe that for every t ≥ 0, H,q,ε

Wt and H,q,ε

t

= q!c(H, q) 0

(

with WuH,ε =

∞ 0

in L 2 (Ω)

(2.23) ⎞

⎛

( Wt

H,q

→ε→0 X t

(

du E(WuH,ε )

)q 2 2

⎜ Hq ⎝ (

WuH,ε E(WuH,ε )2

'

'

⎟ ) 21 ⎠

z 2 −H (1 − sz) H − 2 1z< 1−ε dW (z). s 1

3

Lemma 2.4 The process (WuH,ε , u ∈ [0, T ]) has the same finite dimensional distributions as (YuH,ε , u ∈ [0, T ]), where YuH,ε

=u

H ' − 21

(

∞ 0

'

'

s d B(x) x 2 −H (u − x) H − 2 1x< 1−ε 1

3

where (B(x), x ∈ [0, T ]) is a Brownian motion. Proof Let 0 ≤ u 1 ≤ u 2 ≤ T . Then EWuH,ε WuH,ε 2 ( ∞1 1 1 ' ' 3 ' ' 3 = dzz 2 −H (1 − u 1 z) H − 2 1z< 1−ε z 2 −H (1 − u 2 z) H − 2 1z< 1−ε u1 u2 0 ( ∞ ' 1 ' ' 3 ' 3 = (u 1 u 2 ) H − 2 x 1−2H (u 1 − x) H − 2 (u 2 − x) H − 2 1x 0, λ ∈ R, the function g(u) = 1[0,t] (u)eλu belongs to |H H |. Indeed, for λ ≤ 0, ||g||2|H H |

=

( t( 0

≤

dudveλu eλv |u − v|2H −2

0

( t( 0

t

t

dudv|u − v|2H −2 = (H (2H − 1))−1 t 2H < ∞

0

and for λ > 0, ||g||2|H H | =

( t(

≤e

t

dudveλu eλv |u − v|2H −2

0 0 2λt

(H (2H − 1))−1 t 2H < ∞.

( H,q Consequently, the Hermite-Wiener integral R g(u)d Z u is well-defined. We will see that it is also well-defined in the pathwise sense.

3.3 Wiener Integral in the Riemann-Stieltjes Sense

51

3.3 Wiener Integral in the Riemann-Stieltjes Sense We can define the integral of a deterministic function f with respect to the Hermite process in the Riemann-Stieltjes sense, by using the regularity of the sample paths of the integrator. Let a, b ∈ R such that −∞ < a < b < ∞. Let f : [a, b] → R be a function with bounded variation. Then the Riemann-Stieljes integral of f with (b H,q respect to Z H,q is well-defined (it will be denoted in the sequel by a f (u)d R S Z u ) and it is given by (see Sect. 2.3 in [52]) ( a

b

( H,q

f (u)d R S Z uH,q = f (b)Z b

− f (a)Z aH,q −

b

a

Z uH,q d f (u),

where the integral d f (u) stands for the integral with respect to the bounded variation function f , see Chap. 6 in [36]. The construction of such a Riemann-Stieltjes integral with respect to the Hermite process can be also done on intervals of the form (−∞, b]. Let f : (−∞, b] be a function of bounded variation (i.e. f is with bounded variation on any interval [a ' , b' ] ⊂ (−∞, b] → R) and assume that (

( f (a)Z aH,q

lim

a→−∞

Set

(

b −∞

+

)

b

Z uH,q d f (u)

a

( f (u)d R S Z uH,q = lim

a→−∞ a

Then (see [52], Sect. 2.3) (

(b −∞

b −∞

H,q

f (u)d R S Z u

b

:= L˜ a ∈ R.

f (u)d R S Z uH,q .

(3.8)

(3.9)

is well-defined and H,q

f (u)d R S Z uH,q = f (b)Z b

− L˜ a

(3.10)

In particular, if f : [a, b] → R (−∞ ≤ a < b < ∞) is continuosly differentiable we have ( ( b

a

H,q

f (u)d R S Z uH,q = f (b)Z b

b

− La − a

f ' (u)Z uH,q du

H,q

with L a = limu→a f (u)Z u . The Wiener and Riemann-Stieljes integrals with respect to the Hermite process coincide on the intersection on their domains. (b H,q Proposition 3.2 Let f ∈ |H H | such that a f (u)d R S Z u is well-defined. Then ( a

b

( f (u)d Z uH,q =

a

b

f (u)d R S Z uH,q .

(3.11)

52

3 The Wiener Integral with Respect …

Proof If f is a step function of the form (3.4) (defined via a partition of [a, b]), then both sides of (3.11) coincide (due to construction of the Hermite Wiener integral in Sect. 3.1 and by the definition of the pathwise integral). Then we use the fact that the step functions are dense in |H H |, so for any f ∈ |H H |, there exists a sequence ( f n , n ≥ 1) of step functions such that || f n − f |||H H | →n→∞ 0.

(3.12)

By the isometry of the Hermite-Wiener integral (Proposition 3.1), (( E a

b

)2 ( f n (u) − f (u))d Z uH,q (

= H (2H − 1)

b

(

b

(

a

( ≤ H (2H − 1)

a

b

dudv ( f n (u) − f (u)) ( f n (v) − f (v)) |u − v|2H −2

a b

dudv | f n (u) − f (u)| | f n (v) − f (v)| |u − v|2H −2

a

= || f n − f ||2|H H | →n→∞ 0. On the other hand, (3.12) also implies that f n converges to f almost everywhere (b H,q when n → ∞ and this implies that a f n (u)d R S Z u converges almost surely to (b H,q ∎ a f (u)d R S Z u .

3.4 The Hermite Ornstein-Uhlenbeck Process 3.4.1 Definition and Properties An interesting example of a stochastic process which is defined via a Hermite-Wiener integral is the Ornstein-Uhlenbeck process with respect to the Hermite process, called in the sequel as Hermite Ornstein-Uhlenbeck process. Consider the Langevin equation ( t H,q X s ds + σ Z t , t ≥ 0 (3.13) Xt = ξ − λ 0 H,q with σ, λ > 0 and with initial value ξ ∈ L 2 (Ω). The random is a Hermite ) 1 )noise Z process of order q ≥ 1 with self-similarity index H ∈ 2 , 1 . The unique solution to (3.13) can be written as ( ) ( t −λt λu H,q e d Zu (3.14) ξ+σ , t ≥ 0. Xt = e 0

3.4 The Hermite Ornstein-Uhlenbeck Process

53

(t H,q By Example 3.1, the Wiener integral 0 eλu d Z u is well-defined in L 2 (Ω). Using the expression of the Hermite Wiener integral, we can write, for every t ≥ 0, X t = ξe−λt + σc(H, q) (

t

×

du × e

( d B(y1 )....d B(yq )

Rq q ⊓ −λ(t−u)

0

(u −

) ( − 21 + 1−H q y j )+

j=1

= e−λt ξ + Iq (h t ) with c(H, q) given by (2.5) and with (

t

h t (y! , ..., yq ) = σc(H, q)

du × e

−λ(t−u)

0

q ⊓

(u −

) ( − 21 + 1−H q . y j )+

j=1

When ξ is constant, then the covariance of process X is given by, if s, t ≥ 0, 2 −λ(t+s)

Cov(X t X s ) = σ e

(( E

t

λu

e ( t( 0

= σ 2 α H e−λ(t+s)

0

s

) d Z uH,q

((

s

e

E

λu

0

) d Z uH,q

eλu eλv |u − v|2H −2 dudv.

(3.15)

0

Proposition 3.3 For every T > 0 and for every p ≥ 2, we have sup E|X t | p ≤ C T , p < ∞.

t∈[0,T ]

(3.16)

Proof Assume ξ = 0 for simplicity. By (3.15), we have EX t2

= σ αH

( t(

t

2

0

dudve−λ(t−u) e−λ(t−v) |u − v|2H −2

0

and thus, for every T > 0 ( sup EX t2 ≤ σ 2 α H

t∈[0,T ]

0

T

(

T

dudv|u − v|2H −2 = σ 2 α H T 2H .

0

This bound and the hypercontractivity property (the bound (1.31) in Proposition 1.7) imply (3.16). ∎ Let ξ ∈ R. When q = 1, then X is a Gaussian process and the covariance determines the finite dimensional distributions. When q = 2, then X belongs to the second Wiener chaos and its finite dimensional distributions are given by the cumulants. By using the relation (3.7), we get, since km (X + c) = km (X ) for X ∈ L m (Ω) and c ∈ R,

54

3 The Wiener Integral with Respect …

( ( t ) km (X t ) = km σ e−λ(t−u) d Z uH,q 0 ( )m ( H m−1 = 2 (m − 1)!c(H, q)m β ,1 − H e−mλt du 1 ...du m 2 Rm ×eλ(u 1 +...+u m ) |u 1 − u 2 | H −1 ....|u m−1 − u m | H −1 |u m − u 1H −1 . Due to the Riemann-Stieltjes interpretation, it is possible to give another expression of the Hermite Ornstein-Uhlenbeck process. Proposition 3.4 Let (X t , t ≥ 0) be given by (3.14). Then for every t ≥ 0, Xt = e

−λt

( ξ+

H,q σ Zt

t

− σλ

Z uH,q e−λ(t−u) du.

0

(3.17)

Proof By Proposition 3.2, (

t 0

eλu d Z uH,q =

(

t

eλu d R S Z uH,q .

0

Thus, due to (3.14), Xt = e

−λt

ξ + σe

−λt

= e−λt ξ + σ Z t

(

H,q

H,q Z t eλt

( − σλ

( − σλ

t

)

t

)

Z uH,q du

Z uH,q e−λ(t−u) du.

0

∎ Let us state and prove other properties of the Hermite Ornstein-Uhlenbeck process. They have applications to mathematical finance (see e.g. [20]). Proposition 3.5 Let (X t , t ≥ 0) be given by (3.14). Then, for every t > 0, ( )2 H,q E Xt − X0 − σ Zt →λ→0 0. Proof We have for t > 0, X t − X 0 = ξ(e

−λt

− 1) + σe

−λt

(

t 0

so H,q

Xt − X0 − σ Zt

= ξ(e−λt − 1) +

( 0

t

)

eλu d Z uH,q

) e−λ(t−u) − 1 d Z uH,q .

3.4 The Hermite Ornstein-Uhlenbeck Process

55

Consequently, ( )2 H,q E Xt − X0 − σ Zt (( t )2 ) −λ(t−u) ) e = (Eξ 2 )(e−λt − 1)2 + E − 1 d Z uH,q 0

= (Eξ )(e

−λt

− 1) ( t( t ) −λ(t−u) )) ) e +H (2H − 1) − 1 e−λ(t−v) − 1 |u − v|2H −2 dudv. 2

2

0

0

)

)) ) For every u, we have that e−λ(t−u) − 1 e−λ(t−v) − 1 |u − v|2H −2 1[0,t] (u)1[0,t] (v) converges to zero as λ → 0. It suffices to apply the dominated convergence ∎ theorem. Proposition 3.6 For every T > 0, p ≥ 2, |p | | H,q | sup E |X t − X 0 − σ Z t | →λ→0 0.

t∈[0,T ]

Proof We have by (3.17), X t − X 0 = (e−λt − 1)ξ + σ Z t

H,q

( − σλ 0

t

Z uH,q e−λ(t−u) du

and consequently, |p | | H,q | E |X t − X 0 − σ Z t |

|( | ≤ C p E|e−λt − 1| p + C p λσE || ≤ C p E|e−λt

t 0

( |( | p − 1| + C p λσ E ||

|p | Z uH,q e−λ(t−u) du || t 0

|2 ) 2p | Z uH,q e−λ(t−u) du ||

where we used the hypercontractivity property (1.31). Thus, it suffices to show that |( | sup E ||

t∈[0,T ]

We have, for t ∈ [0, T ],

t 0

|2 |

Z uH,q e−λ(t−u) du ||

< CT .

(3.18)

56

3 The Wiener Integral with Respect …

|( | E ||

t 0

|2 |

Z uH,q e−λ(t−u) du ||

=

( t( 0

=

t

dudve−λ(t−u) e−λ(t−v) EZ uH,q Z vH,q

0

( t( 0

t

dudve−λ(t−u) e−λ(t−v) R H (u, v)

0

( t(

t

dudve−λu e−λv R H (t − u, t − v) ( T( T ≤ T 2H e−λu e−λv ≤ T 2H +2 = C T

=

0

0

0

0

∎

and implies (3.18).

3.4.2 The Stationary Hermite Ornstein-Uhlenbeck Process Let us take the initial value

( ξ=σ

0

eλu d Z uH,q

−∞

in (3.14) and denote, for t ≥ 0, X 0,t = σe−λt

(

t

−∞

eλu d Z uH,q .

(3.19)

To see that this process is well-defined, we notice that, for t > 0, ( 2 EX 0,t = σ 2 H (2H − 1)

t

(

t

−∞ −∞ ( ∞( ∞

dudve−λ(t−u) e−λ(t−v) |u − v|2H −2

e−λu e−λv |u − v|2H −2 ( ∞ ( u 2 −λu = 2σ H (2H − 1) due dveλv |u − v|2H −2 0 0 ( ∞ ( 1 2 −λu 2H −1 = 2σ H (2H − 1) due u dz(1 − z)2H −2 e−λuz 0 0 ( ∞ ≤C u 2H −1 e−λu ≤ C. = σ 2 H (2H − 1)

0

0

0

We show that the law of the process (3.19) is stationary. Recall that a stochastic process (Yt , t ≥ 0) is stationary if and only if (Yt+h , t ≥ 0) has the same finitedimensional distributions as (Yt , t ≥ 0). Proposition 3.7 The process (X 0,t , t ≥ 0) given by (3.19) is stationary.

3.4 The Hermite Ornstein-Uhlenbeck Process

57

Proof For h > 0, X 0,t+h = σe = σe

−λ(t+h)

(

−λ(;t+h)

t+h

−∞ ( t

eλu d Z uH,q e

−∞

λ(u+h)

H,q d Z u+h

= σe

−λt

(

t

−∞

eλu d Z u+h H,q

and since the Hermite process Z H,q has stationary increments, we notice that ((

t

−∞

(( ) H,q eλu d Z u+h , t ≥ 0 ≡(d)

t −∞

) eλu d Z uH,q , t ≥ 0 .

So, (X 0,t+h , t ≥ 0) and (X 0,t , t ≥ 0) have the same finite dimensional distributions. ∎ In particular, the covariance of the process (X 0,t , t ≥ 0) is given by, for 0 ≤ s, t, ( s du dve−λ(t−u) e−λ(s−v) |u − v|2H −2 −∞ −∞ ( ∞( ∞ = α H σ2 dudve−λu e−λv |t − s − (u − v)|2H −2 0 0 ( ∞( ∞ 2 = αH σ dudve−λu e−λv ||t − s| − (u − v)|2H −2 (

t

EX 0,t X 0,s = α H σ 2

0

0

so EX 0,t X 0,s is a function of |t − s|.

Chapter 4

Hermite Sheets and SPDEs

The multiparameter stochastic processes are natural mathematical objects used to modelize the evolution of a system which is not influenced only by the time, but also by other variables, such as the space location. A typical example is the socalled space-time white noise which constitutes the driving noise for many stochastic differential equations. The purpose here is to introduce a multiparameter version of the Hermite processes described in Chap. 2. We mainly follow the lines of Chaps. 2 and 3, but this time in a multidimensional context. We start by properly defining the multiparameter Hermite process (or the Hermite sheet) and then we analyze its main properties. These random fields are elements of the Wiener chaos generated by the Brownian sheet and then they can be expressed as multiple stochastic integrals with respect to this Gaussian field. We prove that the Hermite sheets also enjoy properties like self-similarity, stationarity of the increments or Hölder regularity of the trajectories, but these notions are now understood in a multidimensional manner. We also introduce a Wiener integral with respect to the Hermite sheet, as a counterpart of the integral (3.1) described in Chap. 3. This allows to consider SPDEs driven by the an additive multiparameter Hermite processes. More precisely, one can consider SPDEs of the form H,q Lu(t, x) = Z˙ t,x ,

t ∈ R+ , x ∈ Rd ,

(4.1)

where L is a first or second order operator and Z˙ t,x stands for the formal derivative of the Hermite sheet Z H,q . The mild solution to (4.1) can be expressed as the Wiener integral of the Green kernel associated to the operator L with respect to the Hermite sheet Z H,q . If one takes H,q

Lu(t, x) =

∂ u(t, x) − Δu(t, x), ∂t

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Tudor, Non-Gaussian Selfsimilar Stochastic Processes, SpringerBriefs in Probability and Mathematical Statistics, https://doi.org/10.1007/978-3-031-33772-7_4

59

60

4 Hermite Sheets and SPDEs

(Δ denotes the standard Laplacian operator), then (4.1) is the stochastic heat equation with Hermite noise, while for Lu(t, x) =

∂2 u(t, x) − Δu(t, x), ∂t 2

(4.2)

we obtain the stochastic wave equation. In this chapter, we analyze in details the case of the heat equation driven by a Hermite sheet (other situations have been also treated in the literature, see [12] for the wave equation or [48] for the fractional heat equation). We give a necessary and sufficient condition for the existence of the solution (this condition is in terms of the Hurst parameter of the Hermite noise and of the spatial dimension d) and we study various properties of the law and of paths of this solution.

4.1 Definition of the Hermite Sheet Since we are dealing with multi-indices stochastic process, we will introduce some notation, needed in order to facilitate the lecture. For d ∈ N\ {0} if a = (a1 , a2 , . . . , ad ), b = (b1 , b2 , .., bd ), α = (α1 , .., αd ) are vectors in Rd , we set

ab =

d ∏

ai bi , |a − b|α =

i=1

d ∏

|ai − bi |αi ,

i=1

a/b = (a1 /b1 , a2 /b2 , . . . , ad /bd ), [a, b] =

d d ∏ ∏ [ai , bi ], (a, b) = (ai , bi ), i=1

N ∑ i=0

ai =

N1 ∑ N2 ∑ i 1 =0 i 2 =0

...

Nd ∑

ai1 ,i2 ,...,id , ab =

i d =0

d ∏

aibi ,

i=1

(4.3)

i=1

where N = (N1 , .., Nd ). We use the notation a < b if a1 < b1 , a2 < b2 , . . . , ad < b∏d (analogously for the other inequalities). We write a − n to indicate the product d i=1 (ai − n), if n ∈ R. By β we denote the Beta function (see (2.2)) and we also use the notation d ∏ β(a, b) = β (ai , bi ) i=1

if a = (a1 , .., ad ) and b = (b1 , .., bd ) belong to (0, ∞)d . Let d, q ≥ 1 be integer numbers. The Hermite sheet constitutes a multiparameter version of the process (2.6). Let (W (x), x ∈ Rd ) be a Wiener sheet, characterized by d  its covariance (1.11). Let H = (H1 , . . . , Hd ) ∈ 21 , 1 . The d-parameter Hermite

4.1 Definition of the Hermite Sheet

61

  H,q,d process (or the Hermite sheet) Z t , t ∈ Rd+ of order q ≥ 1 and with Hurst parameter H is defined, for every t ∈ R+ , as H,q,d

Zt

= c(H, q, d)

×

[0,t]

Rdq

(s −

dW (y1 ) . . . dW (yq )

  − 21 + 1−H q y1 )+

. . . (s −

(4.4)

   − 21 + 1−H q yq )+ ds

where x+ = max(x, 0) and ⎛ ⎞ d ∏ H j (2H j − 1) H(2H − 1) 1 1 ⎜ ⎟ c(H, q, d)2 =   =  ⎠ ⎝  q! j=1 β 1 − 1−H j , 2−2H j q q! β 1 − 1−H , 2−2H q 2 q q 2 q q   − 21 + 1−H q y1 )+

(4.5)

If d = 1, we retrieve the constant (2.5). The notation (s − is understood   1−H ∏d − 21 − q j if s = (s1 , ..., sd ) and y1 = (y1,1 , y1,2 , ..., y1,d ). The as j=1 (s j − y1, j ) (q)

constant c(H, q, d) is chosen such that E(Z H,d (t))2 = t 2H for every t ∈ Rd+ , see below. We can also write, for every t ∈ Rd+ , H,q,d

Zt

= Iq (L t )

where Iq stands for the multiple integral of order q with respect to the Wiener sheet and, for y1 , . . . , yq ∈ Rd , the kernel L t (actually, L t = L H,q,d , but we omit these indices in the notation) is given by

L t (y1 , . . . , yq ) = c(H, q, d)

[0,t]

(s −

  − 21 + 1−H q y1 )+

. . . (s −

   − 21 + 1−H q yq )+ ds .

(4.6) Let us see that the Hermite sheet is well-defined and compute its covariance. Lemma 4.7 For every t ∈ [0, ∞)d , the function L t given by (4.6) belongs to L 2 (Rd ). Moreover, for every s, t ∈ Rd+ , we have H,q,d

EZ t

Z sH,q,d =

 1  2H t + s 2H − |t − s|2H := R H (t, s). 2

(4.7)

Proof By Lemma 2.2, for u = (u 1 , ..., u d ), v = (v1 , ..., vd ), a = (a1 , ..., ad ) with ai < − 21 , we notice that

62

4 Hermite Sheets and SPDEs

Rd

⎛ (u − y)a+ (v − y)a+ dy = β(−1 − 2a, a + 1)|u − v|2a+1 ⎝

d ∏

⎞ |u j − v j |2a j +1 ⎠ .

j=1

(4.8) Let us compute the covariance of the random field Z H,q,d . For s = (s1 , ..., sd ), t = (t1 , ..., td ) ∈ Rd+ , we have H,q,d

EZ t

Z sH,q,d = q!c(H, q, d)2 (u −

dy1 ....dyq Rdq   − 21 + 1−H q y1 )+ . . . (u

(v −

  − 21 + 1−H q y1 )+

× ×

[0,t]

[0,s]

−

  − 21 + 1−H q yq )+ du

. . . (v −

  − 21 + 1−H q yq )+ dv

and by using Fubini’s theorem, H,q,d

EZ t

Z sH,q,d = q!c(H, q, d)2  ×

Rd

(u −

du

dv

[0,t] [0,s]   − 21 + 1−H q y)+ (v −

q   − 21 + 1−H q y)+ dy

We compute Rd

(u −

  − 21 + 1−H q y)+ (v

= =

−

  − 21 + 1−H q y)+ dy

   d  1−H 1−H − 21 + q j ∏ − 21 + q j dy1 ....dyd (u j − y j )+ (v j − y j )+ Rd j=1 j=1     1−H 1−H d ∏ − 21 + q j − 21 + q j (v j − y j )+ dy j (u j − y j )+ j=1 R d ∏

and by (4.8), Rd d ∏

(u − 

  1−H − 21 + q j (v y)+

−

  1−H − 21 + q j y)+ dy

 2H j −2 1 1 − H j 2H j − 2 − , |u j − v j | q 2 q q j=1   2H−2 1 1 − H 2H − 2 =β − , |u − v| q . q q 2 =

β

.

4.2 Basic Properties

63

Thus, for s, t ∈ Rd+ ,  H,q,d H,q,d Zs EZ t

1 1 − H 2H − 2 − , 2 q q

q

= q!c(H, q, d) β × du dv|u − v|2H−2 2

[0,t]

[0,s]

 1  2H t + s 2H − |t − s|2H = R H (t, s), = 2 ∎

where we used (4.5).

4.2 Basic Properties We give some immediate distributional and trajectorial properties of the Hermite sheet. We start with the concept of self-similarity for multiparameter stochastic processes. Definition 4.7 Let (X (x), x ∈ Rd ) be a d-parameter stochastic process and let H = (H1 , ..., Hd ) ∈ (0, 1)d . We will say that the process X is self-similar of index H (or H -self-similar) if for every h = (h 1 , ..., h d ) ∈ (0, ∞)d , the stochastic process 

hH X

     x1 xd , ...., , x ∈ Rd = h 1H1 ....h dHd X , x = (x1 , .., xd ) ∈ Rd h h1 hd

x 

has the same finite dimensional distributions as the process X . Proposition 4.23 The Hermite sheet Z H,q,d given by (4.4) is H-self-similar. Proof We have, for h = (h 1 , ..., h d ) ∈ (0, ∞)d and t = (t1 , ..., td ) ∈ [0, ∞)d , H,q,d

hH Z t



h

= c(H, q, d)hH

×

[0,t]

(s −

dW (y1 ) . . . dW (yq )

[0, ht ] Rdq   − 21 + 1−H q y1 )+

= c(H, q, d)hH (h 1 ...h d )−1

 ×

[0,t]

(hs −

− y1 )+

. . . (s −

   − 21 + 1−H q yq )+ ds



1 1−H 2+ q

Rdq 

dW (y1 ) . . . dW (yq )

. . . (hs −

   − 21 + 1−H q yq )+ ds ,

via the change of variables s˜ j = s j h for j = 1, .., d. Now, by using y j = h y˜ j for j = 1, ..., d,

64

4 Hermite Sheets and SPDEs H,q,d

hH Z t

h

= c(H, q, d)hH (h 1 ...h d )−1      − 21 + 1−H − 21 + 1−H q q × (hs − hy1 )+ . . . (hs − hyq )+ ds Rdq

≡(d)

[0,t]

yq y1 dW ( ) . . . dW ( ) h h H,q,d Zt ,

where the last line is obtained via the scaling property of the Hermite sheet.

∎

Recall that the multidimensional increment of a random field is defined by (1.12). Definition 4.8 We say that a d-parameter stochastic process (X (x), x ∈ Rd ) has stationary increments if for every h = (h 1 , ..., h d ) ∈ (0, ∞)d , the d-parameter process   ΔX [h,x+h] , x ∈ Rd has the same finite-dimensional distributions as (X (x), x ∈ Rd ). Proposition 4.24 The Hermite sheet Z H,q,d has stationary increments. Proof The argument is similar to that in the proof of Proposition 2.2.1. For h ∈ (0, ∞)d , H,q,d dW (y1 ) . . . dW (yq ) ΔZ [h,t+h] = c(H, q, d) Rdq

     ×

−

[h,t+h]

(s − y1 )+

1 1−H 2+ q

−

. . . (s − yq )+

1 1−H 2+ q

ds



= c(H, q, d)

×

[0,t]

Rdq

dW (y1 ) . . . dW (yq )

(s + h −

  − 21 + 1−H q y1 )+

. . . (s + h −

   − 21 + 1−H q yq )+ ds

where we made the change of variables u˜ i = u i − h i for i = 1, ..., d. So, H,q,d ΔZ [h,t+h]

= c(H, q, d)

×

[0,t]

Rdq

(s −

dW (y1 + h) . . . dW (yq + h)

  − 21 + 1−H q y1 )+

. . . (s −

   − 21 + 1−H q yq )+ ds

≡(d) Z t

H,q,d

where we used the fact that the Wiener sheet has stationary (multidimensional) increments. ∎

4.3 Wiener Integral with Respect to the Hermite Sheet

65

To analyze the sample paths regularity of the Hermite sheet, we recall the multiparameter version of the Kolmogorov continuity theorem (see e.g. [6]). Theorem 4.4 Let (X t , t ∈ T ) be a d-parameter process, vanishing on the axis, with T a compact subset of Rd . Suppose that there exist constants C, p > 0 and β1 , .., βd > 1 such that | |p β β E |ΔX [t,t+h] | ≤ Ch 1 1 . . . h d d for every h = (h 1 , . . . , h d ) ∈ (0, ∞)d and for every t ∈ T such that t + h ∈ T . Then X admits a continuous modification X˜ . Moreover X˜ has Hölder continuous paths of any orders δ = (δ1 , . . . , δd ) with δi ∈ (0, βi p−1 ) for i = 1, ..., d in the following sense: for every ω ∈ Ω, there exists Cω > 0 such that for every t, t ' ∈ T | | |ΔX [t,t ' ] (ω)| ≤ Cω |t − t ' |δ where |t − t ' |δ = tion (4.3).

∏d j=1

|t j − t 'j |δ j if t = (t1 , . . . , td ), t ' = (t1' , . . . , td' ), see conven-

  H,q,d , t ∈ Rd be a Hermite sheet. Then for every Proposition 4.25 Let Z t s = (s1 , ..., sd ), t = (t1 , ..., td ) ∈ Rd+ , and for every p ≥ 2, |p | | H,q,d | H,q,d p | |t1 − s1 | H1 p . . . |td − sd | Hd p . E |Δ Z [s,t] | = E|Z 1 In particular, the trajectories of the Hermite sheet Z H,q,d are (modulo a modification), Hölder continuous of order δ = (δ1 , ..., δd ) for every δi ∈ (0, Hi ) for i = 1, ..., d. Proof By using the stationarity of the increments and the self-similarity of the Hermite sheet (Propositions 4.23 and 4.24), we can write | | |p |p |p | | | | H,q,d | H,q,d | H,q,d | E |Δ Z [s,t] | = E |ΔZ [0,t−s] | = E |(t − s)H p Z 1 | |p | | H,q,d | = E |(t1 − s1 ) H1 p . . . (td − sd ) Hd p Z 1 | H,q,d p

= E|Z 1

| |t1 − s1 | H1 p . . . |td − sd | Hd p .

The Hölder continuity follows from Theorem 4.4.

∎

4.3 Wiener Integral with Respect to the Hermite Sheet The construction of the Wiener integral with respect to the d-parameter Hermite sheet H,q,d ,t ∈ follows the lines of the one-dimensional case described in Sect. 3.1. Let (Z t Rd ) be a d-parameter Hermite process defined by (4.4) and let E be the set of elementary functions on Rd of the form

66

4 Hermite Sheets and SPDEs

f (t) =

N ∑

ak 1[tk ,tk+1 ) (t)

(4.9)

k=1

with ak ∈ R and 0 ≤ tk < tk+1 for j = 1, ..., N . If f is as in (4.9), then we set f (u)d Z uH,q,d =

Rd

N ∑

H,q,d

ak ΔZ [tk ,tk+1 ] ,

(4.10)

k=1

H,q,d

where ΔZ [tk ,tk+1 ] stands for the high order increment over the interval [tk , tk+1 ], see (1.12). Let us introduce the transfer operator ( J f )(y1 , ..., yq ) = c(H, q, d)

Rd

f (u)

q ∏

(u −

  − 21 + 1−H q y j )+ du,

(4.11)

j=1

for y1 , ..., yq ∈ Rd , with c(H, q, d) given by (4.5). Using the convention (4.3), this can be written as (J f )(y1 , ..., yq ) = c(H, q, d)

Rd

f (u 1 , ..., u d )

q ∏ d ∏

− (u k − y j,k )+



1 1−Hk 2+ q



du 1 ...du d

j=1 k=1

if y j = (y j,1 , ..., y j,d ) for j = 1, ..., q. By (4.10) and (1.12), we can write Rd

f (u)d Z uH,q,d =

N ∑ k=1

=

N ∑ k=1

H,q,d

ak ΔZ [tk ,tk+1 ] ⎛ ak ⎝

∑

⎞ ∑d d− i=1 ri

(−1)

H,q,d Z tk+r1 ,1 ,...,tk+rd ,d ⎠

r ∈{0,1}d

if tk = (tk,1 , ...., tk,d ) for k = 1, ..., N + 1. By the definition of the transfer operator (4.11), we get

=

Rd N ∑

f (u)d Z uH,q,d



=

N ∑ k=1

∑d

(−1)d−

i=1 ri

r ∈{0,1}d

k=1

×

∑

ak

Rdq

ak

  J 1[0,tk+r1 ,1 ]×...[0,tk+rd ,d ] (y1 , ..., yq )dW (y1 )....dW (yq ) Rdq

  J 1[t1,k ,t1,k+1 ] × .... × 1[td,k ,td,k+1 ] (y1 , ..., yq )dW (y1 )....dW (yq )

4.3 Wiener Integral with Respect to the Hermite Sheet

67

=

Rdq

(J f )(y1 , ..., yq )dW (y1 )....dW (yq ).

This motivates the following definition: let f : Rd → R be a measurable function such that f ∈ HH,d , i.e. Rdq

(J f )2 (y1 , ..., yq )dy1 ....dyq < ∞.

Then we set by definition

Rd

f (u)d Z uH,q,d :=

Rdq

(J f )(y1 , ..., yq )dW (y1 )....dW (yq )

(4.12)

with J f given by (4.11). This will be called the Wiener integral of f with respect to Hermite sheet Z H,q,d , or, as in the one-dimensional case, the Hermite-Wiener integral. In particular, this integral is well defined if | | f ∈ |HH,d | | | where |HH,d | is the set of measurable function f : Rd → R such that

||

f ||2|HH,d |

:=

Rd

Rd

| f (u)| · | f (v)||u − v|2H−2 dudv < ∞.

(4.13)

| | The set |HH,d | is is a Banach space with respect to the norm || · |||HH,d | defined by (4.13) and we have the following inclusions (see [32]), | | 1 L 2 (Rd ∩ L 1 (Rd )) ⊂ L H (Rd ) ⊂ |HH,d | ⊂ HH,d . The Wiener integral with respect to the Hermite sheet is an isometry, Indeed, assume that f is a step function as in (4.9). Then 2

 E Rd

=

N ∑

f (u)d Z uH,q,d H,q,d

H,q,d

ak al EΔZ [tk ,tk+1 ] ΔZ [tl ,tl+1 ]

k,l=1

=

N ∑

ak al

k,l=1

=

N ∑ k,l=1

∑

∑d

(−1)d−

i=1 ri

∑ r∈{0,1}d

∑d

(−1)d−

j=1

ρj

H,q,d

H,q,d

EZ tk+r Z tl+ρ

ρ∈{0,1}d

r∈{0,1}d

ak al

∑

∑d

(−1)d−

i=1 ri

∑ ρ∈{0,1}d

∑d

(−1)d−

j=1

ρj

RH (tk+r , tl+ρ )

68

4 Hermite Sheets and SPDEs

with R H given by (4.7). By Lemma 2.1, 

2

E Rd

=

f (u)d Z uH,q,d

N ∑

ak al H1 (2H1 k,l=1 tk+1,d tk+1,1 ...

tk,1

×|u 1 − v1 |

tk,d 2H1 −2

− 1) . . . Hd (2Hd − 1)

tl+1,1

du 1 ...du d



tl,1 2Hd −2

. . . |u d − vd |

tl+1,d

...

dv1 ...dvd

tl,d

.

(4.14)

| | Let us introduce the following notation, for f, g ∈ |HH,d |, ⟨ f, g⟩HH,d = H(2H − 1)

Rd



and || f ||2HH,d = H(2H − 1)

Rd

f (u)g(v)|u − v|2H−2 dudv

Rd

Rd

f (u) f (v)|u − v|2H−2 dudv.

Then, by (4.14), 2

 E Rd

=

N ∑

f (u)d Z uH,q,d

ak al ⟨1[tk,1 ,tk+1,1 ]×...×[tk,d ,tk+1,d ] , 1[tl,1 ,tl+1,1 ]×...×[tl,d ,tl+1,d ] ⟩HH,d

k,l=1

=

N ∑

ak al ⟨1[tk ,tk+1 ] , 1[tk ,tk+1 ] ⟩HH,d

k,l=1

= ⟨ f, f ⟩HH,d . | | | | Since | the| set E is dense in HH,d (see [|12, 32]), | the above isometry can be extended to |HH,d |. Therefore, for every f, g ∈ |HH,d | we will have  

 E Rd

f (u)d Z uH,q,d

 Rd

g(u)d Z uH,q,d

= ⟨ f, g⟩HH,d .

(4.15)

4.4 The Stochastic Heat Equation with Hermite Noise

69

4.4 The Stochastic Heat Equation with Hermite Noise The construction of the Wiener integral with respect to a Hermite sheet allows to consider SPDEs (stochastic partial differential equations) with Hermite noise. We will illustrate here the case of the stochastic heat equation (but other types of SPDEs can also be treated, see e.g. [12] for the wave equation with Hermite noise). The simplest deterministic heat equation in an open set U ⊂ Rd (d ≥ 1) can be written as ∂u = Δu(t, x), t ≥ 0, x ∈ U (4.16) ∂t where Δ is the standard Laplacian operator, i.e. if F : U ⊂ Rd → R, then ΔF(x) =

d ∑ ∂2 F

∂xi2

i=1

(x), if x = (x1 , ..., xd ) ∈ U.

The Eq. (4.16) describes the heat flow in an homogeneous medium, u(t, x) modelling the temperature at time t and at the point x in space. The stochastic heat equation constitutes a model for the heat flow under a random perturbation. In most situations, the random perturbation (or the random noise) is assumed to be a Gaussian field, the most prominent example being the space-time white noise (see e.g. [14] or [51]). Here we will analyze the stochastic heat equation with a non-Gaussian noise, namely the Hermite noise. More precisely, we consider the stochastic partial differential equation ∂ (H ,H),q,d+1 u(t, x) = Δu(t, x) + Z˙ t,x 0 , ∂t

t ≥ 0, x ∈ Rd

(4.17)

with vanishing initial condition u(0, x) = 0 for every x ∈ Rd . In (4.17), Z˙ (H0 ,H),q,d+1 stands for the formal derivative of 

(H0 ,H),q,d+1

Zt

 , t ∈ Rd+1 ,

which is a d + 1-parameter Hermite sheet, of order q ≥ 1, with Hurst parameter d+1  d  , where H = (H1 , ..., Hd ) ∈ 21 , 1 . (H0 , H) ∈ 21 , 1

70

4 Hermite Sheets and SPDEs

4.4.1 Existence of the Solution Let us introduce the kernel  G(t, x) =

  2 (2πt)−d/2 exp − ||x|| if t > 0, x ∈ Rd , 2t , 0 if t ≤ 0, x ∈ Rd

(4.18)

where || · || denotes the Euclidean norm in Rd . That is, G is the Green kernel (or the − Δu = 0. fundamental solution) associated to nthe heat equation, i.e it solves ∂u ∂t To define the solution to (4.17), we start with the following remark: if f ∈ C 1,2 (R+ × R), set v(t, x) =

ds 0

Then it is possible to prove equation ∂ v(t, x) = ∂t



t

Rd

dyG(t − s, x − y) f (s, y).

(4.19)

that v defined by (4.19) solves the partial differential 1 Δv(t, x) + f (t, x), 2

t ≥ 0, x ∈ Rd .

This motivates the definition of the solution to (4.17): we replace f (s, y)dsdy by (H ,H),q,d+1 (H ,H),q,d+1 dsdy which is interpreted as d Z s,y0 , the Wiener integral with Z˙ s,y0 respect to the Hermite sheet introduced above in Sect. 4.3. More exactly, one defines the mild solution to (4.17) by u(t, x) =

t 0

Rd

(H0 ,H),q,d+1 G(t − s, x − y)d Z s,y ,

t ≥ 0, x ∈ Rd ,

(4.20)

where the above integral is a Hermite-Wiener integral. Using the definition (4.12) of the Hermite-Wiener integral, we cal also write u(t, x) = Iq (Ht,x ) where Iq is the multiple stochastic integral with respect to the Wiener sheet W and for s1 ...., sq ∈ R+ and z 1 , ..., z q ∈ Rd ,   Ht,x (s1 , z 1 ), ..., (sq , z q ) = c(H, q, d) (u − s1 )





t

du

Rd 0   1−H − 21 + q 0

(y − z 1 )

  − 21 + 1−H q

dyG(t − u, x − y)

. . . (u − sq )

. . . (y − z q )

  1−H − 21 + q 0

  − 21 + 1−H q

. (4.21)

Let S(Rd ) be the Schwarz space of rapidly decreasing C ∞ -functions on Rd . For f ∈ S(Rd ), we define its Fourier transform by

4.4 The Stochastic Heat Equation with Hermite Noise

71

(F f )(ξ) =

ξ ∈ Rd ,

ei⟨x,ξ⟩ d x, Rd

with ⟨·, ·⟩ standing for the Euclidean scalar product in Rd . The Fourier transform of f is also well-defined when f ∈ L 1 (Rd ) or f ∈ L 2 (Rd ). We will use the Parseval identity αH

Rd

Rd

ϕ(x)ψ(y)|x − y|

2H−2

d yd x = aH

Rd

Fϕ(ξ)Fψ(ξ)|ξ|1−2H dξ (4.22)

with some positive constant aH , for every ϕ, ψ such that

Rd

Rd

|ϕ(x)ψ(y)||x − y|2H−2 d yd x < ∞.

We recall that the quantities |ξ|1−2H or |x − y|2H−2 are understood via the convention (4.3). Let us give a necessary and sufficient condition for the existence of the mild solution (4.20). Proposition 4.26 Let T > 0 and let t ∈ [0, T ], x ∈ Rd be fixed. Then the mapping (s, y) → G(t − s, x − y)1(0,t) (s) | | belongs to |H(H0 ,H),d+1 | if and only if d < 4H0 +

d ∑

(2Hi − 1).

(4.23)

i=1

In this case, sup Eu(t, x)2 < ∞.

t∈[0,T ]

Proof We have, by using the isometry of the Hermite-Wiener integral (4.15) Eu(t, x)2 = ||G(t − ·, x − ∗)1(0,t) (·)||2HH,d+1 t t = α H0 du dv|u − v|2H0 −2 0

0

dydz Rd

Rd

G(t − u, x − y)G(t − v, x − z) f (y − z) where G is defined by (4.18) and f (y − z) =

d ∏ i=1

Hi (2Hi − 1)|yi − z i |2Hi −2 = H(2H − 1)|y − z|2H−2 .

72

4 Hermite Sheets and SPDEs

We recall that the Fourier transform of the Green kernel (4.18) with respect to the spatial variable is given by F G(t, ·)(ξ) = e−

t||ξ||2 2

.

(4.24)

By using (4.24) and the Parseval relation (4.22), we obtain, with c(H0 , H) a generic constant that may change, Eu(t, x)2 = c(H0 , H) = c(H0 , H)

t

t

dv|u − v|2H0 −2

du 0

0

t

t

dv|u − v|2H0 −2

du 0

0

Rd



Rd

where |ξ|1−2H =

dξF G(t − u, ·)(ξ)F G(t − u, ·)(ξ)|ξ|1−2H dξe−

d ∏

(t−u)||ξ||2 2

e−

(t−v)||ξ||2 2

|ξ|1−2H

|ξ j |1−2H j .

j=1

Therefore, by the change of variables Eu(t, x)2





t

= c(H0 , H)

t

du

0



t

= c(H0 , H) 0



t

= c(H0 , H)

t

2H0 −2

dv|u − v|

t

dξe− 2 (u+v)||ξ|| 1

2

Rd

d ∏

|ξ j |1−2H j

j=1

(u + v)

− d2

(u + v)

H1 +...+Hd −d



dξe− 2 ||ξ|| 1

2

Rd

dv|u − v|2H0 −2 (u + v)− 2 (u + v) H1 +...+Hd −d , d

0

where we used the fact that the integral of variables uv = z in the integral dv, Eu(t, x)2



0

du 0

dv|u − v|2H0 −2

0

du

√ u + vξi =  ξi for i = 1, ..., d, we find





t

u



dξe− 2 ||ξ|| is finite. Next, by the change 1

Rd

2

dv|u − v|2H0 −2 (u + v)− 2 (u + v) H1 +...+Hd −d 0 0 1 t = c(H0 , H) du × u 2H0 −1 u H1 +...+Hd −d dz(1 − z)2H0 −2 (1 + z) H1 +...+Hd −d 0 0 t = c(H0 , H) du × u 2H0 −1 u H1 +...+Hd −d = c(H0 , H)

du

0

d

4.4 The Stochastic Heat Equation with Hermite Noise

73

1 since the integral 0 dz(1 − z)2H0 −2 (1 + z) H1 +...+Hd −d is finite for H0 > 21 . We t finally notice that the integral 0 duu 2H0 −1 u H1 +...+Hd −d converges if and only if ∎ d < 2H0 + H1 + ... + Hd , and this leads to (4.23).

4.4.2 Self-similarity We prove that the solution (4.20) is self-similar with respect to its temporal variable. Proposition 4.27 Assume (4.23) and let x ∈ Rd be fixed. Then the process (u(t, x), t ≥ 0) is self-similar of order γ = H0 −

H1 + ... + Hd d + . 2 2

(4.25)

Proof Let a > 0. By (4.21), u(at, x)





= c(H, q, d) (u − s1 )

W (ds1 , dz 1 )....W (dsq , dz q )

Rdq   1−H 1 − 2+ q 0

(y − z 1 )

  − 21 + 1−H q

. . . (u − sq )

  1−H − 21 + q 0



at

du 0

Rd

dyG(at − u, x − y)

  − 1 + 1−H

. . . (y − z q ) 2 q = c(H, q, d)a ... W (ds1 , dz 1 ) . . . W (dsq , dz q )

R



t

Rd   − 21 + 1−H q

(y − z 1 )

= c(H, q, d)a

t

(y −

Rd

R

  1−H − 21 + q 0 s1 )+

. . . (au −

  1−H − 21 + q 0 sq )+

  − 21 + 1−H q

. . . (y − z q ) ... W (d(as1 ), dz 1 ) . . . W (d(asq ), dz q )

Rd

R

Rd

dyG α (a(t − u), x − y)(au − Rd     − 21 + 1−H − 21 + 1−H q q . . . (y − z q )+ . z 1 )+

du 0

R

dyG(a(t − u), x − y)(au −

du 0



Rd

  1−H − 21 + q 0 as1 )+

. . . (au −

  1−H − 21 + q 0 asq )+

Using the fact that the Brownian sheet W is 21 -self-similar with respect to its time variable and its increments are stationary in space, we get (we denote, as usual, by ≡(d) the equivalence of the finite dimensional distributions),

74

4 Hermite Sheets and SPDEs

u(at, x) ≡(d) c(H, q, d)aa

−q



1 1−H0 2+ q



q





W (ds1 , dz 1 ) . . . W (dsq , dz q ) R Rd     t 1−H 1−H − 21 + q 0 − 21 + q 0 . . . (u − sq )+ du d yG(a(t − u), −y)(u − s1 )+ Rd 0     − 21 + 1−H − 21 + 1−H q q . . . (y − z q )+ . (y − z 1 )+ a2

R Rd

...

We use the following property of the Green kernel G(at, x) = a − 2 G(t, a − 2 x). d

1

Then d

u(at, x) ≡(d) c(H, q, d)a H0 a − 2 t

du





R Rd

...

R Rd

W (ds1 , dz 1 ) . . . W (dsq , dz q )

    1−H 1−H − 21 + q 0 − 21 + q 0 1 − dyG(t − u, −a 2 y)(u − s1 )+ . . . (u − sq )+

Rd     − 21 + 1−H − 21 + 1−H q q (y − z 1 )+ . . . (y − z q )+ . 0

Next, we make the change of variables a − 2 yi = y˜i for i = 1, ..., d and then z˜ i = 1 a − 2 z i for i = 1, ..., d. We will obtain 1

u(at, x) ≡(d) c(H, q, d)a H0 a − 2 a 2 d



d



t

du Rd

0 1

(a 2 (y −

R

Rd

dyG(t − u, −y)(u −

  − 21 + 1−H 1 q z 1 ))+ . . . (a 2 (y   1−Hi q ∑d 1 H0 − 2 i=1 2 + q

...

1

. . . (u −

  1−H − 21 + q 0 sq )+

  − 21 + 1−H q z q ))+

= c(H, q, d)a a 1 1 ... W (ds1 , d(a 2 z 1 )) . . . W (dsq , d(a 2 z q ))

R Rd t

du 0

(y −



R

Rd

dyG(t − u, −y)(u −

Rd   − 21 + 1−H q z 1 )+

. . . (y −

  1−H − 21 + q 0 s1 )+

  − 21 + 1−H q z q )+ .

1

W (ds1 , d(a 2 z 1 )) . . . W (dsq , d(a 2 z q ))

R   1−H − 21 + q 0 s1 )+

−

Rd

. . . (u −

  1−H − 21 + q 0 sq )+

4.4 The Stochastic Heat Equation with Hermite Noise

75

Finally, we use the spatial scaling property of the Brownian sheet W to get −

∑d  1

q

+

1−Hi

 dq

u(at, x) ≡(d) c(H, q, d)a H0 a 2a i=1 2 q a 4 ... W (ds1 , z 1 ) . . . W (dsq , z q ) R Rd R Rd t     1−H0 1−H0 − 1+ − 1+ du dyG(t − u, −y)(u − s1 ) 2 q . . . (u − sq ) 2 q Rd   − 21 + 1−H q

0

(y − z 1 )

. . . (y − z q )

  − 21 + 1−H q

so for x ∈ R fixed, (u(at, x), t ≥ 0) ≡(d)

  ∑d d i=1 Hi a H0 − 2 + 2 u(t, x), t ≥ 0 = (a γ u(t, x), t ≥ 0)

with γ in (4.25).

∎

Remark 4.7 Notice that the self-similarity index γ given by (4.25) is strictly positive, due to condition (4.23). Also, we have γ < H0 < 1 since γ = H0 −

d−

∑d i=1

Hi

2

≤ H0 < 1.

4.4.3 Regularity of Sample Paths Let us study the regularity of the mapping t → u(t, x), with x ∈ Rd fixed. To this end, we start by analyzing the temporal increment of the solution. Proposition 4.28 Assume (4.23). Then for every 0 ≤ s < t and for every x ∈ Rd , E |u(t, x) − u(s, x)|2 ≤ C|t − s|2γ ,

(4.26)

with γ given by (4.25) and with C > 0 not depending on s, t, x. Proof We write, for 0 ≤ s < t and for x ∈ Rd , u(t, x) − u(s, x) =

t s

Rd s



+ 0

G(t − u, x − y)d Z H,q,d

Rd

H,q,d (G(t − u, x − y) − G(s − u, x − y)) d Z u,y

=: T1 (s, t) + T2 (s, t). The two terms T1 (s, t) and T2 (s, t) from above will be estimated separately. First, we have, by the isometry (3.6) and (4.22),

76

4 Hermite Sheets and SPDEs

E|T1 (s, t)|2 t t = α H0 dudv|u − v|2H0 −2 s s dydzG(t − u, x − y)G(t − v, x − y)|y − z|2H−2 Rd

Rd

= c(H0 , H)

t s



t

dudv|u − v|



0

t−s

dξe− 2 (t−u+s−v)||ξ|| 1

dudv|u − v|2H0 −2

0

We use the change of variables ξ˜ j =

t−s

E|T1 (s, t)|2 = c(H0 , H) ×





0



√

t−s

d ∏

|ξ j |1−2H j

j=1



dξe− 2 (u+v)||ξ|| 1

2

Rd

d ∏

|ξ j |1−2H j .

j=1

u + vξ j for j = 1, ..., d to get dudv|u − v|2H0 −2 (u + v)−d+H1 +...+Hd

0

dξe− 2 ||ξ||

Rd

2

Rd

s t−s

= c(H0 , H)

2H0 −2

d ∏

1

2

t−s



|ξ j |1−2H j

j=1 t−s

dudv|u − v|2H0 −2 (u + v)−d+H1 +...+Hd 1 t−s = c(H0 , H) dvu 2γ−1 dz(1 − z)2H0 −2 (1 + z) H1 +...+Hd −d = c(H0 , H)

0

0

0

0

= c(H0 , H)|t − s|

2γ

where γ is defined by (4.25). Next, E|T2 (s, t)|2 s s 2H0 −2 = c(H0 , H) dudv|u − v| dξ|ξ|1−2H d R 0 0  1   2 − 2 (t−u)|ξ|2 − 21 (s−u)|ξ|2 −(t−v)|ξ|2 −e − e−(s−v)|ξ| e e s s t−s t−s = c(H0 , H)|t − s|2H0 dudv|u − v|2H0 −2 dξ|ξ|1−2H 0 0 Rd  1  1  1 1 2 2 2 2 e− 2 (t−s)(1+u)|ξ| − e− 2 (t−s)u|ξ| e− 2 (t−s)(1+v)|ξ| − e− 2 (t−s)v|ξ| , v˜ = where we performed the change of variables u˜ = s−u t−s 1 ˜ set ξi = (t − s) 2 ξi for every i = 1, ..., d. In this way,

s−v . t−s

in the next step, we

4.4 The Stochastic Heat Equation with Hermite Noise

E|T2 (s, t)|2 ∑d − d2 + i=1 (1−2Hi )



s t−s



s t−s

77

2H0 −2

= C|t − s| (t − s) dudv|u − v| 0 0   1 1 1 2 2 2 e− 2 (2+u+v)|ξ| − 2e− 2 (1+u+v)|ξ| + e− 2 (u+v)|ξ| − 2 ∞ ∞ 2γ 2H0 −2 ≤ C(t − s) dudv|u − v| dξ|ξ|1−2H 0 0 Rd   1 1 1 2 2 2 e− 2 (2+u+v)|ξ| − 2e− 2 (1+u+v)|ξ| + e− 2 (u+v)|ξ| ∞ ∞ 1 2 dudv|u − v|2H0 −2 dξ|ξ|1−2H e− 2 |ξ| = C(t − s)2γ Rd 0 0  1 ∑d d 1 ∑d d (2 + u + v)− 2 − 2 i=1 (1−2Hi ) − 2(1 + u + v)− 2 − 2 i=1 (1−2Hi )  d 1 ∑d +(u + v)− 2 − 2 i=1 (1−2Hi ) ∞ ∞ 2γ dudv|u − v|2H0 −2 = C(t − s) 0 0   (2 + u + v)2γ−2H0 − 2(1 + u + v)2γ−2H0 + (u + v)2γ−2H0 . 2H0

dξ|ξ|1−2H Rd

For u, v close to infinity, we have (2 + u + v)2γ−2H0 − 2(1 + u + v)2γ−2H0 + (u + v)2γ−2H0 ≤ C|u + v|2γ−2H0 −2 and the above integral dudv is convergent at infinity because 2γ < 2 or equivalently   d ∑ 1 Hi ≤ 1. H0 − d− 4 i=1

∎

As an immediate consequence of Proposition 4.28, we obtain the Hölder regularity in time of the solution. Corollary 4.2 The mapping t → u(t, x) is Hölder continuous of order δ for every δ ∈ (0, γ), with γ given by (4.25). Proof From (4.26) and by using the hypercontractivity (1.31), we have for p ≥ 2, E |u(t, x) − u(s, x)| p ≤ C p |t − s| pγ for every 0 ≤ s < t and x ∈ Rd . The conclusion follows by Kolmogorov’s continuity criterion. ∎

78

4 Hermite Sheets and SPDEs

4.4.4 A Decomposition Theorem Consider the random field (u(t, x), t ≥ 0, x ∈ Rd ) which is mild solution to (4.17) and assume (4.23) is satisfied. We have seen in Proposition 4.27 that the the random field u is self-similar in time and it is pretty obvious that it has no stationary increments with respect to the time variable. The purpose is to give a decomposition of the solution to the fractional stochastic heat equation (4.17) as a sum of a self-similar process in time with temporal stationary increments and of another process with very nice sample paths in time. This decomposition is useful, among others, to get the behavior of p-variation of the solution and to estimate the drift parameter of the stochastic heat equation. We will use the so-called pinned string method introduced in [25] and then used by several authors (in e.g. [21, 26, 38, 45]). Let us set, for t ≥ 0, x ∈ Rd , U (t, x) =

R

(H0 ,H),q,d+1 . (G((t − s)+ , x − y) − G((−s)+ , x − y)) d Z s,y

Rd

(4.27)

This is called the pinned string process. We can also write U (t, x) =

t 0

+



R 0

(H0 ,H),q,d+1 G(t − s, x − y)d Z s,y (H0 ,H),q,d+1 . (G(t − s, x − y) − G(−s, x − y)) d Z s,y

−∞

R

We will first show that U has the scaling property in time and it also has stationary temporal increments. Then we will show that the difference u(t, x) − U (t, x) has C ∞ -sample paths with respect to the time variable. Proposition 4.29 Let x ∈ Rd be fixed. Then the process (U (t, x), t ≥ 0) defined by (4.27) is γ-self-similar and it has stationary increments. Proof The self-similarity follows as above in the proof of Proposition 4.27. Let us show that for every h > 0, the stochastic processes (U (t + h, x) − U (h, x), t ≥ 0) and (U (t, x)), t ≥ 0) have the same finite dimensional distributions. We can write, for h > 0, U (t + h, x) − U (h, x) (H0 ,H),q,d+1 = (G((t + h − s)+ , x − y) − G((h − s)+ , x − y)) d Z s,y R Rd (H0 ,H),q,d+1 (d) ≡ (G((t − s)+ , x − y) − G((−s)+ , x − y)) d Z s,y R

Rd

= U (t, x)

4.4 The Stochastic Heat Equation with Hermite Noise

79

where we used the fact that the process (Z (H0 ,H),q,d+1 (t, x), t ≥ 0) has stationary increments in time. ∎ From the above result, it follows (see e.g. [46]) that the covariance of the process (U (t, x), t ≥ 0) is given by  EU (1, x)2  2γ t + s 2γ − |t − s|2γ , 2

EU (t, x)U (s, x) =

t, s ≥ 0.

On the other hand, since (U (t, x), t ≥ 0) is not a Gaussian process, the covariance did not determine the probability distribution of this stochastic process (except when d = 1). Set, for t > 0 and x ∈ Rd , Y (t, x) = u(t, x) − U (t, x) so Y (t, x) = −



0

−∞

Rd

(H0 ,H),q,d+1 . (G(t − s, x − y) − G(−s, x − y)) d Z s,y

(4.28)

We will show that the random field Y has smooth sample paths with respect to the time variable. Proposition 4.30 Let (Y (t, x), t > 0, x ∈ Rd ) be given by (4.28) and assume (4.23). Then for every x ∈ Rd , the sample paths t → Y (t, x) are absolutely continuous and of class C ∞ on (0, ∞). Proof Set

'

Y (t, x) = −



0 −∞

Rd

∂ (H0 ,H),q,d+1 G(t − s, x − y)d Z s,y , ∂t

t > 0, x ∈ Rd , (4.29)

the formal derivative of Y with respect to the time variable t. We will have (again C > 0 denotes a generic constant) via (4.22) '



E|Y (t, x)| = C  2

0



0

2H0 −2



dudv|u − v| dξ|ξ|1−2H Rd   ∂ −(t−v)||ξ||2 ∂ −(t−u)||ξ||2 e e ∂t ∂t 0 0 =C dudv|u − v|2H0 −2 dξ|ξ|1−2H +4 −∞

−∞

Rd

−∞ −∞ −(t−u)||ξ||2 −(t−v)||ξ||2

×e =C

∞ t



t

∞

e

2H0 −2

dudv|u − v|



dξ|ξ|5−2H e−(u+v)||ξ|| Rd

2

80

4 Hermite Sheets and SPDEs

and by setting (u + v) 2 ξi = ξ˜i for i = 1, ..., d, we get 1



'

E|Y (t, x)| = C

∞

2

t



∞

dudv|u − v|2H0 −2 (u + v)− 2 (u + v)− 2 d

1

∑d

i=1 (5−2Hi )

t

dξ|ξ|1−2H +2α e−|ξ| Rd ∞ ∞ d 1 ∑d =C dudv|u − v|2H0 −2 (u + v)− 2 (u + v)− 2 i=1 (5−2Hi ) . 2

t

Thus, with z =

t

u v

E|Y ' (t, x)|2 ∞ ∞ ∑d 2d =C dv du(u − v)2H0 −2 (u + v)− 2 −2d+ i=1 Hi v ∞ t ∞ ∑d 2d 2 ∑d 2H0 −1− 2d 2 −2d+ i=1 Hi dz(z − 1)2H0 −1 (1 + z)− α −2d+ α i=1 Hi =C dvv 1 ∞ t ∞ 2γ−1−2d 2H0 −2 =C dvv (1 − z) (1 + z)2γ−2H0 −2d . t

1

The integral dv is finite because 2γ − 2d < 0 (see Remark 4.7) and the integral d x is finite at 1 because 2H0 > 1 and at infinity because 2γ − 2d − 1 < 0 (again by Remark 4.7). Therefore (Y ' (t, x), t > 0) is a well defined random field and consequently t → Y (t, x) is absolute continuous and of class C 1 on (0, ∞). Similarly (see [26, 38] or [45] for details), we can deal with the nth derivative and we can show ∎ that t → Y (t, x) is of class C ∞ on (0, ∞).

4.4.5

p-Variation

We will use the above decomposition theorem in order to obtain the p-variation in time of the solution u. Let us first define the concept of p-variation. Consider 0 ≤ A1 < A2 two real numbers and let ti = A1 +

i ( A2 − A1 ), N

i = 0, ..., N

(4.30)

be a partition of the interval [ A1 , A2 ]. Let (v(t, x), t ≥ 0, x ∈ Rd ) a general random field and define, for x ∈ Rd , p > 0 and N ≥ 1 N,p S[A1 , A2 ] (v(·, x))

=

N −1 ∑ i=0

|v(ti+1 , x) − v(ti , x)| p .

(4.31)

4.4 The Stochastic Heat Equation with Hermite Noise

81

We will say  that v admits a temporal  p-variation over the interval [A1 , A2 ] if the N,p sequence S[A1 ,A2 ] (v(·, x)), N ≥ 1 converges in probability as N → ∞. For the solution to the fractional stochastic heat equation, we have the following result. Recall that γ is given by (4.25). Theorem 4.5 Let (u(t, x), t ≥ 0, x ∈ Rd ) be defined by (4.20) and assume (4.23). Then 1 N, 1 S[A1γ, A2 ] (u(·, x)) → N →∞ E |U (1, 0)| γ ( A2 − A1 ) in probability where U is given by (4.27). Proof In a first step, we will show that the p-variation of the random field Y given by (4.28) vanishes, for every p ≥ γ1 . Indeed, for p ≥ γ1 > 1 N −1 ∑

|Y (ti+1 , x) − Y (ti , x)| p

i=0

≤

sup |a−b|≤

A2 −A1 N

|Y (a, x) − Y (b, x)| p−1

N −1 ∑

|Y (ti+1 , x) − Y (ti , x)| .

i=0

Recall from Proposition 4.30 that Y has absolute continuous temporal sample paths. The continuity of Y with respect to the time variable and p > 1 implies that sup |a−b|≤

A2 −A1 N

|Y (a, x) − Y (b, x)| p−1 → N →∞ 0

∑ N −1 |Y (ti+1 , x) − Y (ti , x)| is bounded by the total pointwise while the quantity i=0 variation of t → Y (t, x) over the interval [ A1 , A2 ]. Consequently, N −1 ∑

|Y (ti+1 , x) − Y (ti , x)| p → N →∞ 0 pointwise.

(4.32)

i=0

Let us now analyze the γ1 -variation of the random field U . We have, by the scaling property in time of U obtained in Proposition 4.29, if “=(d) ” stands for the equality in distribution, N −1 ∑

|U (ti+1 , x) − U (ti , x)| p =(d) (A2 − A1 )γ p

i=0

= (A2 − A1 )γ p

N −1 1 ∑ |U (i + 1, x) − U (i, x)| p n γ p i=0

1 VN n γ p−1

(4.33)

82

4 Hermite Sheets and SPDEs

with VN =

N −1 1 ∑ |U (i + 1, x) − U (i, x)| p . N i=0

The sequence (U (i + 1, x) − U (i, x), i ≥ 0) is stationary due to the fact that (U (t, x), t ≥ 0) has stationary increments, see Proposition 4.29. On the other hand, U (t, x) is an element of the qth Wiener chaos, for every t ≥ 0, x ∈ Rd . Actually, we have   U (t, x) = G t,x (s1 , z 1 ), . . . , (sq , z q ) W (ds1 , dz 1 ) . . . W (dsq , dz q ) R

Rd

where   G t,x (s1 , z 1 ), . . . (sq , z q ) = c(H, q) du dy (G((t − u)+ , x − y) − G((−u)+ , x − y)) R

×(u − ×(y −

Rd     1−H 1−H − 21 + q 0 − 21 + q 0 s 1 )+ . . . (u − sq )+     − 21 + 1−H − 21 + 1−H q q . . . (y − z q )+ . z 1 )+

Moreover,     G t,x (s1 , z 1 ), . . . (sq , z q ) − G s,x (s1 , z 1 ), . . . (sq , z q ) = c(H, q) du dy (G((t − u)+ , x − y) − G((s − u)+ , x − y)) R

×(u − ×(y −

Rd     1−H0 1−H 1 − 2+ q − 21 + q 0 s1 )+ . . . (u − sq )+     − 21 + 1−H − 21 + 1−H q q . . . (y − z q )+ z 1 )+



= c(H, q) ×(u − ×(y −



dy (G((t − s − u)+ , x − y) − G((−u)+ , x Rd     1−H 1−H − 21 + q 0 − 21 + q 0 (s1 − s))+ . . . (u − (sq − s)+     1−H 1 − 2+ q − 21 + 1−H q . . . (y − z q )+ z 1 )+ du

R

− y))

so     G t,x (s1 , z 1 ), . . . (sq , z q ) − G s,x (s1 , z 1 ), . . . (sq , z q )   = G t−s,x (s1 − s, z 1 ), . . . (sq − s, z q ) .

(4.34)

4.4 The Stochastic Heat Equation with Hermite Noise

83

Since the kernel G t,x satisfies the shifting property (4.34), it follows from Theorem 8.3.1 in [37] that the sequence (U (i + 1, x) − U (i, x), i ≥ 0) is also mixing. Therefore (see e.g. Chap. 2 in [37]) VN → N →∞ E |U (1, 0)| p almost surely and in L 1 (Ω).

(4.35)

By (4.33) and (4.35) we obtain (since the convergence in law to a constant implies the convergence in probability) N −1 ∑

|U (ti+1 , x) − U (ti , x)| p → N →∞

i=0

⎧ 1 ⎪ ⎨0, if p > γ1 E|U (1, 0)| γ ( A2 − A1 ) if p = ⎪ ⎩ +∞ if p < γ1

1 γ

(4.36)

in probability. Now, by using Minkovski’s inequality  N −1 ∑

 1p |U (ti+1 , x) − U (ti , x)| p

i=0 N,p ≤ S[A1 , A2 ] (u(·, x)))

≤

 N −1 ∑ i=0

|U (ti+1 , x) − U (ti , x)| p

−

 N −1 ∑

 1p |Y (ti+1 , x) − Y (ti , x)| p

i=0

 1p +

 N −1 ∑

 1p |Y (ti+1 , x) − Y (ti , x)| p

.

i=0

To get the conclusion, it suffices to use (4.32) and (4.36).

∎

Remark 4.8 From the proof of Theorem 4.5, we can notice that the solution (4.20) has zero p-variation in time on any interval [ A1 , A2 ] if p > γ1 .

Chapter 5

Statistical Inference for Stochastic (Partial) Differential Equations with Hermite Noise

While the statistical inference for SDEs and SPDEs driven by the Brownian motion or, more general, by a Gaussian noise, has a long history, the statistical inference for systems driven by Hermite processes and sheets is at its beginning. Our aim is to illustrate how certain parameters can be estimated in two situations where the Hermite processes appear as random perturbations. The first example concerns the Hermite Ornstein-Uhlenbeck process introduced and analyzed in Sect. 3.4. We recal that this stochastic process is the solution to the Langevin equation with Hermite noise (3.13). In (3.13), we will assume that the initial value ξ vanishes (for simplicity) and the drift coefficient is λ = 1 (the estimation of this parameter has been done in [28]), so our observed process solves the equation {

t

Xt = −

H,q

X s ds + σ Z t

, t ≥ 0.

(5.1)

0

The purpose is to estimate the remaining two parameters, i.e. the Hurst index H and the volatility coefficient σ, from the onservation of the solution to (5.1) at the discrete times ti = Ni , i − 0, 1, ..., N along the unit interval [0, 1]. Our estimators will be expressed in terms of the quadratic (and generalized) variation of the Hermite Ornstein-Uhlenbeck process. Therefore, we need to analyze the asymptotic behavior of these random sequences, i.e. the sequences VN (X ) given by (5.7) and the 1 -variation defined by the left-hand side of (5.31). To this end, we notice that the H solution to the Langevin equation (5.1) can be written as the sum of the noise term (the Hermite process) and of another more regular stochastic process, and we show the asymptotic behavior of the power variation of the solution is given by the power variation of the noise term, which has already been studied in Chap. 2. Then, we construct in a standard way, estimators for the parameters H and σ and we deduce the asymptotic properties of these estimators. This has been essentially treated in [2]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Tudor, Non-Gaussian Selfsimilar Stochastic Processes, SpringerBriefs in Probability and Mathematical Statistics, https://doi.org/10.1007/978-3-031-33772-7_5

85

86

5 Statistical Inference for Stochastic (Partial) Differential Equations …

In the second example we deal with the stochastic heat equation driven by the Hermite sheet from Sect. 4.4. Here we illustrate how the drift parameter θ appearing in the SPDE (5.33) can be estimated. The approach is also based on the analysis of the temporal p- variation of the mild solution u θ to (5.33), i.e. the of the sequence N −1 ∑

|u θ (ti+1 , x) − u θ (ti , x)| p ,

i=0

where ti =

i ,i N

= 0, 1, ..., N and x ∈ Rd is fixed.

5.1 Parameter Identification for the Hermite Ornstein-Uhlenbeck Process Let us consider the Hermite Ornstein-Uhlenbeck (HOU) process given by (3.14) with ξ = 0 and λ = 1. That is, {

X t = σe−t

t 0

eu d Z uH,q .

(5.2)

We can also write, due to Definition 3.1, X t = Iq (h t ), where, for t ≥ 0 and y1 , ..., yq ∈ R, {

t

h t (y1 , .., yq ) = σc(H, q)

e

−(t−u)

0

q ∏

(u −

) ( − 21 + 1−H q yi )+ du,

(5.3)

i=1

where c(H, q) is the constant given by (2.5). Via the Langevin equation (3.13), we have the decomposition, for t ≥ 0, {

t

Xt = −

H,q

X s ds + σ Z t

H,q

= Yt + σ Z t

(5.4)

0

{

with

t

Yt = −

X s ds, 0

t ≥ 0.

(5.5)

5.1 Parameter Identification for the Hermite Ornstein-Uhlenbeck Process

87

5.1.1 Quadratic Variation Our purpose is to use the p- variation of the process (5.2) in order to identify its Hurst and diffusion parameters. Let us consider the partition of the unit interval [0, 1] given by i for i = 0, ..., N and for N ≥ 1 (5.6) ti = N We will assume that the process X is observed at times ti and we will define estimators for the parameters H and σ in (5.4) based on these observations. Define, for every N ≥ 1, the sequence of (centered and renormalized) quadratic variations ⎡ N −1 ∑

(

)2

⎤

X ti+1 − X ti ⎥ ( )2 − 1⎦ H,q H,q i=0 σ 2 E Z ti+1 − Z ti ] N −1 [ )2 1 ∑ N 2H ( = X − X − 1 . ti+1 ti N i=0 σ 2

VN (X ) =

1 N

⎢ ⎣

(5.7)

We aim at finding the limit behavior, as N → ∞, of the sequence VN (X ). We will benefit from the behavior of the quadratic variation of the noise of (5.4) which is well-known: while for q = 1, this is the famous Breuer-Major theorem (see e.g. [8, 27]), for q ≥ 2, it has been obtained in [11, 49]. Let us recall these results. By “→(d) ” we denote the convergence in distribution and by N (0, 1) we indicate the standard normal law. ( ) Theorem 5.6 Assume H ∈ 21 , 1 and q ≥ 1 integer. Let VN (Z H,q ) be given by ⎤ ⎡ ( )2 H,q H,q N −1 − Z Z ti+1 ti 1 ∑⎢ ⎥ (5.8) VN (Z H,q ) = ⎣ ( )2 − 1⎦ . N i=0 H,q H,q E Z ti+1 − Z ti Then 1. If q = 1 and H ∈

(1 2

) , 34 , √ K 1,1 N VN (Z H,1 ) →(d) N →∞ N (0, 1).

(5.9)

2. If q ≥ 2 or q = 1 and H > 43 , then Kq N

2−2H q

'

VN (Z H,q ) → N →∞ Z 1H ,2 in L 2 (Ω)

(5.10)

88

5 Statistical Inference for Stochastic (Partial) Differential Equations … '

where Z 1H ,2 is a Rosenblatt random variable with Hurst parameter H ' = 2(Hq−1) + 1. The constants K 1,1 , K i , i = 1, .., q are explicit, we refer to [11, 27] for their expression. Actually the result at point 1. above holds for every H ∈ (0, 34 ) (and even for H = 43 under a different renormalization) but this case will be not discussed here. We will prove in the sequel that VN (X ) keeps the same behavior as the quadratic variations of the noise of (5.4). This means that the drift process (Yt , t ∈ [0, T ]) given by (5.5) does not affect the behavior of VN (X ). This is due to the regularity of Y and to the fact that the correlation between the increments of Y and Z H,q is weak enough. Proposition 5.31 Let VN (X ) be given by (5.7). ( ) ' 1. Assume H ∈ 21 , 1 and q ≥ 2. Then, with K q , Z 1H ,2 from (5.10) Kq N 2. If q = 1 and H ∈

(1 2

2−2H q

'

VN (X ) → N →∞ Z 1H ,2 in L 2 (Ω).

) , 34 , √ K 1,1 N VN (X ) →(d) N →∞ N (0, 1).

with K 1,1 from (5.9). Proof Assume first q ≥ 2. We decompose VN (X ) as follows: VN (X ) =

] N −1 [ N −1 ( )2 )2 1 1 ∑ 2H ( 1 ∑ H,q H,q Yti+1 − Yti −1 + 2 N N 2H Z ti+1 − Z ti σ N i=0 N i=0 +

N −1 ) ) ( H,q 2 1 ∑ 2H ( (q,H ) Yti+1 − Yti Z ti+1 − Z ti N σ N i=0

= VN (Z (q,H ) ) + T1,N + T2,N with

(5.11) N −1

)2 1 2H −1 ∑ ( Yti+1 − Yti N 2 σ i=0

(5.12)

) ) ( H,q 2 2H −1 ∑ ( H,q Yti+1 − Yti Z ti+1 − Z ti N . σ i=0

(5.13)

T1,N = and

N −1

T2,N =

In (5.11), the limit of the sequence VN (Z H,q ) is known from Theorem 5.6. We will prove that the other terms does not contribute to the limit. That is, we show that, for i = 1, 2 and for every p ≥ 1

5.1 Parameter Identification for the Hermite Ornstein-Uhlenbeck Process

N

2−2H q

89

Ti,N → N →∞ 0 in L p (Ω).

(5.14)

The summand T1,N can be easily estimated, by using Hölder’s inequality. Indeed, for every p ≥ 1,

E|T1,N | = σ p

−2 p

N

(2H −1) p

| N −1 ({ )2 || p |∑ ti+1 | | E| X s ds | | | ti i=0

≤ σ −2 p N (2H −1) p N ( p−1) E

N −1 ({ ∑

N −1 { ∑ i=0

)2 p

ti+1

X s ds

ti

i=0

≤ σ −2 p N (2H −1) p N − p E

(5.15)

ti+1

|X s |2 p ds ≤ C N (2H −2) p (5.16)

ti

where we used (3.16). Consequently (5.14) holds true for i = 1, since for every p≥1 |p | 2−2H 1 | | E |N q T1,N | ≤ C N (2H −2)(1− q ) p → N →∞ 0. To analyze the term T2,N , we need to use its Wiener chaos decomposition (actually, a direct proof based on Hölder inequality can be done only for q ≥ 3). We can write, for every N ≥ 1 and for every i = 0, ..., N , Yti+1 − Yti = Iq (h i,N ) where h i,N = h ti+1 − h ti (h t is given in (5.3)), i.e. {

{

ti+1

h i,N (y1 , . . . , yq ) = σd(q, H )

ds

H,q

due

−(s−u)

0

ti

and

s

H,q

Z ti+1 − Z ti

q ∏

−( 21 + 1−H q )

(u − yl )+

(5.17)

l=1

= Iq (li,N )

with (recall that L t = L H,q is the kernel of the Hermite process, see (2.4)) li,N (y1 , . . . , yq ) = L ti+1 (y1 , . . . , yq ) − L ti (y1 , . . . , yq )

(5.18)

for every y1 , ..., yq ∈ R. In this way, via the product formula for multiple integrals (1.33),

90

5 Statistical Inference for Stochastic (Partial) Differential Equations … N −1

T2,N =

2 2H −1 ∑ N Iq (h i,N )Iq (li,N ) σ i=0 N −1 q

=

∑ (r ) 2 2H −1 ∑ ∑ N r !(Cqr )2 I2q−2r (h i,N ⊗r li,N ) := T2,N σ r =0 i=0 r =0 q

with, for r = 0, .., q, (r ) T2,N

( )2 ∑ N −1 2 2H −1 r r! I2q−2r (h i,N ⊗r li,N ). = N q i=0 σ

(5.19)

We will obtain (5.14) for i = 2 if we show that 2−2H q

E|N

(r ) 2 T2,N | → N →∞ 0

(5.20)

via the hypercontractivity property (1.31). Assume r = q. Notice that h i,N , li,N are symmetric functions. Below, we denote by c, C generic strictly positive constants not depending on N and that are allowed to change from one line to another. We have N −1 ∑ (q) T2,N = cN 2H −1 ⟨h i,N , li,N ⟩ L 2 (Rq ) i=0 (q)

with h i,N , li,N given by (5.17), (5.18) respectively. Thus T2,N is a deterministic sequence and (q)

|T2,N | = cN 2H −1

N −1 { ∑ i=0

∏ q

×

{

Rq

dy1 ...dyq

{

ti+1

ti

−( 21 + 1−H q

(u − y j )+

)

{

due−(s−u)

0

∏ q

ti+1

dv ti

j=1

s

ds

−( 21 + 1−H q )

(v − y j )+

.

j=1

By using the formula (2.3) in Lemma 2.2, we obtain (q) |T2,N |

= cN

2H −1

N −1 { ∑ i=0

≤ cN

2H −1

ti

s

ds

due

−(s−u)

0

ti

N −1 { ∑ i=0

{

ti+1

{

ti+1

{

ti+1

du 0

ti+1

dv|u − v|2H −2

ti 1

ds

{

dv|u − v|2H −2 ≤ C N 2H −2 (5.21)

ti

and (5.20) holds for r = q because we assumed q ≥ 2. Now assume 1 ≤ r ≤ q − 1. We study the sequence

5.1 Parameter Identification for the Hermite Ornstein-Uhlenbeck Process (r ) T2,N

= cN

2H −1

I2q−2r

( N −1 ∑

91

) h i,N ⊗r li,N

i=0

where, again via (2.3) and Fubini, (h i,N ⊗r li,N )(y1 , ..., y2q−2r ) { ti+1 { s N −1 { ti+1 ∑ (2H −2) qr = cN 2H −1 ds due−(s−u) dv|u − v| 0

ti

i=0

−( 21 + 1−H q

(u − y1 )+

)

ti

−( 21 + 1−H q

...(u − yq−r )+

−( 21 + 1−H q )

×(v − yq−r +1 )+

) −( 21 + 1−H q )

...(v − y2q−2r )+

for every y1 , ..., y2q−2r ∈ R. By isometry (see (3.6)) | | | (r ) |2 E |T2,N |

(5.22)

= cN 4H −2 || = C N 4H −2

N −1 ∑ i=0 N −1 ∑

N −1 ∑

˜ r li,N ||2 2 2q−2r ≤ cN 4H −2 || h i,N ⊗ L (R )

i=0

h i,N ⊗r li,N ||2L 2 (R2q−2r )

⟨h i,N ⊗r li,N , h j,N ⊗r l j,N ⟩ L 2 (R2q−2r )

i, j=0

= C N 4H −2

N −1 { ti+1 ∑

due−(s−u)

0

i, j=0 ti

(2H −2) qr

×|u − v|

{ s ds

|u ' − v ' |

{ ti+1

{ t j+1 dv

ti

(2H −2) qr

|u − u ' |

ds '

tj

{ s'

'

0

(2H −2) (q−r) q

'

du ' e−(s −u )

{ t j+1

dv '

tj

|v − v ' |

) (2H −2) (q−r q

(5.23)

where we used again (5.23). Now, we majorize the exponential function by 1 and the integral over [0, s] × [0, s ' ] by the integral over [0, 1]2 . We will obtain, for every r = 1, ..., q − 1, N −1 { | | ∑ | (r ) |2 E |T2,N | ≤ C N 4H −2 i, j=0

{

1

1

du 0

0

du '

{

{

ti+1

dv ti

t j+1

dv '

tj

×|u − v|(2H −2) q |u ' − v ' |(2H −2) q |u − u ' |(2H −2) r

r

(q−r ) q

|v − v ' |(2H −2)

≤ C N 4H −4

(q−r ) q

(5.24)

and consequently (5.20) holds since (see e.g. [11]) {

(2H −2) qr

[0,1]4

dudu ' dvdv ' |u − v|

(2H −2) qr

|u ' − v ' |

) (2H −2) (q−r q

|u − u ' |

) (2H −2) (q−r q

|v − v ' |