Stochastic Modelling and Applied Probability

Transcription

1 Stochastic Mechanics Random Media Signal Processing and Image Synthesis Mathematical Economics and Finance Stochastic Optimization Stochastic Control Stochastic Models in Life Sciences Edited by Advisory Board Stochastic Modelling and Applied Probability (Formerly: Applications of Mathematics) 67 B. Rozovskiĭ P.W. Glynn M. Hairer I. Karatzas F.P. Kelly A. Kyprianou Y. Le Jan B. Øksendal G. Papanicolaou E. Pardoux E. Perkins H.M. Soner

2 For further volumes:

3 Jean Jacod Philip Protter Discretization of Processes

4 Jean Jacod Institut de Mathématiques Université Paris VI Pierre et Marie Curie Place Jussieu 4 Paris Cedex, Paris France jean.jacod@upmc.fr Philip Protter Department of Statistics Columbia University Amsterdam Av New York, NY USA pep2117@columbia.edu Managing Editors Boris Rozovskiĭ Division of Applied Mathematics Brown University 182 George St Providence, RI USA rozovsky@dam.brown.edu Peter W. Glynn Institute of Computational and Mathematical Engineering Stanford University Via Ortega 475 Stanford, CA USA glynn@stanford.edu ISSN Stochastic Modelling and Applied Probability ISBN e-isbn DOI / Springer Heidelberg Dordrecht London New York Library of Congress Control Number: Mathematics Subject Classification (2010): 60F05, 60G44, 60H10, 60H35, 60J75, 60G51, 60G57, 60H05, 60J65, 60J25 Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Preface Two decades ago the authors of this book undertook the study of the errors one makes when numerically approximating the solutions of stochastic differential equations driven by Lévy processes. In particular we were interested in the normalized asymptotic errors of approximations via an Euler scheme, and it turned out we needed sophisticated laws of large numbers and central limit theorems that did not yet exist. While developing such tools, it became apparent that they would be useful in a wide range of applications. One usually explains the difference between probability and statistics as being that probability theory lays the basis for a family of models, and statistics uses data to infer which member or members of that family best fit the data. Often this reduces to parameter estimation, and estimators are shown to be consistent via a Law of Large Numbers (LLN), and the accuracy of an estimator is determined using a Central Limit Theorem (CLT), when possible. The case of stochastic processes, and even stochastic dynamical systems, is of course more difficult, since often one is no longer estimating just a parameter, but rather one is estimating a stochastic process, or worse trying to tell which family of models actually does fit the data. Examples include using data to determine whether or not a model governing a dynamical system has continuous paths or has jumps, or trying to determine the dimension of the driving Brownian forces in a system of stochastic differential equations. This subject, broadly speaking, is a very old subject, especially as concerns asymptotic studies when the time parameter tends to infinity. The novelty presented here in this book is a systematic study of the case where the time interval is fixed and compact (also known as the finite horizon case). Even in the finite horizon case however, efforts predate the authors study of numerical methods for stochastic differential equations, and go back 5 years earlier to attempts to find the volatility coefficient of an Itô process, via a fine analysis of its quadratic variation, by the first author joint with Valentine Genon-Catalot. This in turn builds on the earlier work of G. Dohnal, which itself builds on earlier work; it is indeed an old yet still interesting subject. There are different variations of LLNs and CLTs one might use to study such questions, and over the last two decades substantial progress has been made in finding such results, and also in applying them via data to delve further into the v

6 vi Preface unknown, and to reveal structures governing complicated stochastic systems. The most common examples used in recent times are those of financial models, but these ideas can be used in models of biological, chemical, and electrical applications as well. In this book we establish, in a systematic way, many of the recent results. The ensuing theorems are often complicated both to state, and especially to prove, and the technical level of the book is (inevitably, it seems) quite demanding. This is a theory book, and we do not treat applications, although we do reference papers that use these kinds of results for applications, and we do indicate at the end of most chapters how this theory can be used for applications. An introduction explaining our approach, and an outline of how we have organized the book, can be found in the Introductory Chapter 1. In addition, in Chap. 1 we present several sketches of frameworks for potential applications of our theory, and indeed, these frameworks have inspired much of the development of the theory we present in this book. If one were to trace back how we came to be interested in this theory, the history would have to center on the work and personality of Denis Talay and his équipe at INRIA in Sophia-Antipolis, as well as that of Jean Mémin at the University of Rennes. Both of these researchers influenced our taste in problems in enduring ways. We would also like to thank our many collaborators in this area over the years, with a special mention to Tom Kurtz, whose work with the second author started this whole enterprise in earnest, and also to Yacine Aït-Sahalia, who has provided a wealth of motivations through applications to economics. We also wish to thank O.E. Barndorff-Nielsen, S. Delattre, J. Douglas, Jr., V. Genon-Catalot, S.E. Graversen, T. Hayashi, Yingying Li, Jin Ma, S. Méléard, P. Mykland, M. Podolskij, J. San Martin, N. Shephard, V. Todorov, S. Torres, M. Vetter, and N. Yoshida, as well as A. Diop, for his careful reading of an earlier version of the manuscript. The authors wish to thank Hadda and Diane for their forbearance and support during the several years involved in the writing of this book. The second author wishes to thank the Fulbright Foundation for its support for a one semester visit to Paris, and the National Science Foundation, whose continual grant support has made this trans-atlantic collaboration possible. Paris, France New York, USA Jean Jacod Philip Protter

7 Contents Part I Introduction and Preliminary Material 1 Introduction ContentandOrganizationoftheBook When X isabrownianmotion The Normalized Functionals V n (f, X) The Non-normalized Functionals V n (f, X) When X isabrownianmotionplusdrift The Normalized Functionals V n (f, X) The Non-normalized Functionals V n (f, X) When X is a Brownian Motion Plus Drift Plus a Compound Poisson Process TheLawofLargeNumbers The Central Limit Theorem Some Prerequisites Semimartingales First Decompositions and the Basic Properties ofasemimartingale Second Decomposition and Characteristics ofasemimartingale A Fundamental Example: Lévy Processes ItôSemimartingales SomeEstimatesforItôSemimartingales Estimates for Bigger Filtrations The Lenglart Domination Property Limit Theorems Stable Convergence in Law Convergence for Processes Criteria for Convergence of Processes Triangular Arrays: Asymptotic Negligibility vii

8 viii Contents Convergence in Law of Triangular Arrays Bibliographical Notes Part II The Basic Results 3 Laws of Large Numbers: The Basic Results Discretization Schemes Semimartingales with p-summablejumps Law of Large Numbers Without Normalization TheResults TheProofs LawofLargeNumberswithNormalization PreliminaryComments TheResults TheProofs Applications Estimation of the Volatility DetectionofJumps Bibliographical Notes Central Limit Theorems: Technical Tools Processes with F-Conditionally Independent Increments The Continuous Case The Discontinuous Case TheMixedCase Stable Convergence Result in the Continuous Case A Stable Convergence Result in the Discontinuous Case AnApplicationtoItôSemimartingales The Localization Procedure A Stable Convergence for Itô Semimartingales Central Limit Theorems: The Basic Results The Central Limit Theorem for Functionals Without Normalization The Central Limit Theorem, Without Normalization Proof of the Central Limit Theorem, Without Normalization The Central Limit Theorem for Normalized Functionals: Centering with Conditional Expectations StatementoftheResults TheProof The Central Limit Theorem for the Processes V n (f, X) AssumptionsandResults LocalizationandEliminationofJumps Proof of the Central Limit Theorem for V n (f, X) The Central Limit Theorem for Quadratic Variation A Joint Central Limit Theorem Applications Estimation of the Volatility...176

9 Contents ix DetectionofJumps Euler Schemes for Stochastic Differential Equations Bibliographical Notes Integrated Discretization Error StatementsoftheResults Preliminaries AnApplicationofItô sformula Reduction of the Problem Proof of the Theorems Proof of Theorem Proof of Theorem Proof of Theorem Proof of Theorem Part III More Laws of Large Numbers 7 First Extension: Random Weights Introduction The Laws of Large Numbers for V n (F, X) The Laws of Large Numbers for V n (F, X) ApplicationtoSomeParametricStatisticalProblems Second Extension: Functions of Several Increments Introduction The Law of Large Numbers for V n (F, X) and V n (F, X) The Law of Large Numbers for V n (Φ, k n,x) The LLN for V n (F, X), V n (F, X) and V n (Φ, k n,x) TheResults TheProofs Applications to Volatility Third Extension: Truncated Functionals ApproximationforJumps Approximation for the Continuous Part of X Local Approximation for the Continuous Part of X:PartI From Local Approximation to Global Approximation Local Approximation for the Continuous Part of X:PartII Applications to Volatility Part IV Extensions of the Central Limit Theorems 10 The Central Limit Theorem for Random Weights Functionals of Non-normalized Increments Part I Functionals of Non-normalized Increments Part II Functionals of Normalized Increments ApplicationtoParametricEstimation Bibliographical Notes...296

10 x Contents 11 The Central Limit Theorem for Functions of a Finite Number of Increments Functionals of Non-normalized Increments TheResults An Auxiliary Stable Convergence Proof of Theorem Functionals of Normalized Increments TheResults EliminationofJumps Preliminaries for the Continuous Case The Processes Y n and Y n Proof of Lemma Joint Central Limit Theorems Applications Multipower Variations and Volatility SumsofPowersofJumps DetectionofJumps Bibliographical Notes The Central Limit Theorem for Functions of an Increasing Number of Increments Functionals of Non-normalized Increments TheResults An Auxiliary Stable Convergence Result Proof of Theorem Functionals of Normalized Increments TheResults PreliminariesfortheProof Proof of Lemma Block Splitting Proof of Lemma The Central Limit Theorem for Truncated Functionals A Central Limit Theorem for Approximating the Jumps Central Limit Theorem for Approximating the Continuous Part TheResults Proofs Central Limit Theorem for the Local Approximation of the Continuous Part of X StatementsofResults Elimination of the Jumps and of the Truncation The Scheme of the Proof in the Continuous Case Proof of Lemma Proof of Lemma Proof of Theorem Another Central Limit Theorem Using Approximations of the Spot Volatility...410

11 Contents xi StatementsofResults Proofs Application to Volatility Bibliographical Notes Part V Various Extensions 14 Irregular Discretization Schemes Restricted Discretization Schemes Law of Large Numbers for Normalized Functionals Central Limit Theorem for Normalized Functionals TheResults Preliminaries The Scheme of the Proof when X is Continuous Proof of Lemma Proof of Lemma Application to Volatility Bibliographical Notes Higher Order Limit Theorems Examples of Degenerate Situations Functionals of Non-normalized Increments Applications Bibliographical Notes Semimartingales Contaminated by Noise Structure of the Noise and the Pre-averaging Scheme StructureoftheNoise The Pre-averaging Scheme Law of Large Numbers for General (Noisy) Semimartingales Central Limit Theorem for Functionals of Non-normalized Increments TheResults A Local Stable Convergence Result A Global Stable Convergence Result Proof of Theorem Laws of Large Numbers for Normalized Functionals and Truncated Functionals StatementofResults TheProofs Laws of Large Numbers and Central Limit Theorems for Integral Power Functionals TheLawsofLargeNumbers Central Limit Theorems: The Results SomeEstimates Proof of Theorem The Quadratic Variation Bibliographical Notes...563

12 xii Contents Appendix A.1 EstimatesforItôSemimartingales A.2 Convergence of Processes A.3 Triangular Arrays A.4 ProcessesofFiniteVariation A.5 SomeResultsonLévyProcesses Assumptions on the Process X References Index of Functionals Index...593

13 Basic Notation R = the set of real numbers, R + =[0, ) Q = the set of rational numbers, Q + = Q R + N = the set {0, 1,...} of natural integers, N ={1, 2,...} Z = the set of relative integers R d = the Euclidean d-dimensional space; the components of x R d are (x 1,...,x d ) R d = the Borel σ -field of R d, R = R 1 M + d d = the set of nonnegative symmetric d d matrices M ++ d d = the set of nonnegative symmetric invertible d d matrices x = the absolute value of x R, x = Euclidean norm of x R d d(x,b) = the distance between a point x and a subset B, in a metric space y = the transpose of the vector or matrix y [x] = the integer part of x R (biggest n Z such that n x) a b = sup(a, b), a b = inf(a, b), ifa,b R x + = x 0, x = ( x) 0, if x R 1 A = the indicator function of the set A A c = the complement of the set A ε a = the Dirac measure sitting at point a δ ij = the Kronecker symbol, equal to 1 if i = j and to 0 otherwise Notation for convergences: a.s. a.s. (almost sure) convergence for random variables P L L-s convergence in probability for random variables convergence in law for random variables stable convergence in law for random variables a.s. = a.s. convergence for processes, for Skorokhod s topology P = convergence in probability for processes, for Skorokhod s topology u.c.p. = convergence in probability for processes, for the local uniform topology xiii

14 xiv Basic Notation L = convergence in law for processes, for Skorokhod s topology L-s = stable convergence in law for processes, for Skorokhod s topology Miscellaneous: f(x)= o(g(x)) as x x 0 if f(x)/g(x) 0asx x 0 f(x)= O(g(x)) as x x 0 if lim sup x x0 f(x)/g(x) < càdlàg = right-continuous with left limits càglàd = left-continuous with right limits u n v n means that both sequences u n /v n and v n /u n are bounded