An Introduction to Continuous-Time Stochastic Processes_ Theory, Models, and Applications to Finance

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Unformatted text preview: Modeling and Simulation in Science, Engineering and Technology Vincenzo Capasso David Bakstein An Introduction to ContinuousTime Stochastic Processes Theory, Models, and Applications to Finance, Biology, and Medicine Third Edition Modeling and Simulation in Science, Engineering and Technology Series Editor Nicola Bellomo Politecnico di Torino Torino, Italy Editorial Advisory Board K. Aoki Kyoto University Kyoto, Japan K.J. Bathe Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, MA, USA M. Chaplain Division of Mathematics University of Dundee Dundee, Scotland, UK P. Degond Department of Mathematics, Imperial College London, London, United Kingdom P. Koumoutsakos Computational Science & Engineering Laboratory ETH Zürich Zürich, Switzerland H.G. Othmer Department of Mathematics University of Minnesota Minneapolis, MN, USA K.R. Rajagopal Department of Mechanical Engineering Texas A&M University College Station, TX, USA Y. Tao Dong Hua University Shanghai, China A. Deutsch Center for Information Services and High-Performance Computing Technische Universität Dresden Dresden, Germany T.E. Tezduyar Department of Mechanical Engineering & Materials Science Rice University Houston, TX, USA M.A. Herrero Departamento de Matematica Aplicada Universidad Complutense de Madrid Madrid, Spain A. Tosin Istituto per le Applicazioni del Calcolo “M. Picone” Consiglio Nazionale delle Ricerche Roma, Italy More information about this series at Vincenzo Capasso • David Bakstein An Introduction to Continuous-Time Stochastic Processes Theory, Models, and Applications to Finance, Biology, and Medicine Third Edition Vincenzo Capasso ADAMSS (Interdisciplinary Centre for Advanced Applied Mathematical and Statistical Sciences) Universit`a degli Studi di Milano Milan, Italy David Bakstein ADAMSS (Interdisciplinary Centre for Advanced Applied Mathematical and Statistical Sciences) Universit`a degli Studi di Milano Milan, Italy ISSN 2164-3679 ISSN 2164-3725 (electronic) Modeling and Simulation in Science, Engineering and Technology ISBN 978-1-4939-2756-2 ISBN 978-1-4939-2757-9 (eBook) DOI 10.1007/978-1-4939-2757-9 Library of Congress Control Number: 2015938721 Mathematics Subject Classification (2010): 60-01, 60FXX, 60GXX, 60G07, 60G10, 60G15, 60G22, 60G44, 60G51, 60G52, 60G57, 60H05, 60H10, 60H30, 60J25, 60J35, 60J60, 60J65, 60K35, 91GXX, 92BXX, 93E05, 93E15 Springer New York Heidelberg Dordrecht London Springer Science+Business Media New York 2005, 2012, 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. © Printed on acid-free paper Springer Science+Business Media LLC New York is part of Springer Science+Business Media (www. springer.com) Preface to the Third Edition In this third edition, we have included additional material for use in modern applications of stochastic calculus in finance and biology; in particular, Chap. 5 on stability and ergodicity is completely new. We have thought that this is an important addition for all those who use stochastic models in their applications. The sections on infinitely divisible distributions and stable laws in Chap. 1, random measures, and L´evy processes in Chap. 2, Itˆo–L´evy calculus in Chap. 3, and Chap. 4, have been completely revisited. Fractional calculus has gained a significant additional room, as requested by various applications. The Karhunen-Lo`eve expansion has been added in Chap. 2, as a useful mathematical tool for dealing with stochastic processes in statistics and in numerical analysis. Various new examples and exercises have been added throughout the volume in order to guide the reader in the applications of the theory. The bibliography has been updated and significantly extended. We have also made an effort to improve the presentation of parts already included in the previous editions, and we have corrected various misprints and errors made aware of by colleagues and students during class use of the book in the intervening years. We are very grateful to all those who helped us in detecting them and suggested possible improvements. We are very grateful to Giacomo Aletti, Enea Bongiorno, Daniela Morale, and the many students for checking the final proofs and suggesting valuable changes. Among these, the PhD students Stefano Belloni and Sven Stodtmann at the University of Heidelberg deserve particular credit. Kevin Payne and (as usual) Livio Pizzocchero have been precious for bibliographical references, and advice. v vi Preface to the Third Edition Enea Bongiorno deserves once again special mention for his accurate final editing of the book. We wish to pay our gratitude to Avner Friedman for having allowed us to grasp many concepts and ideas, if not pieces, from his vast volume of publications. Allen Mann from Birkh¨ auser in New York deserves acknowledgement for encouraging the preparation of this third edition. Last but not least, we acknowledge the precious editorial work of the many (without specific names) at Birkh¨ auser, who have participated in the preparation of the book. Most of the preparation of this third edition has been carried out during the stays of VC at the Heidelberg University (which he wishes to acknowledge for support by BIOMS, IWR, and the local HGS), and at the “Carlos III” University in Madrid (which he wishes to thank for having offered him a Chair of Excellence there). Milan, Italy Milan, Italy Vincenzo Capasso David Bakstein Preface to the Second Edition In this second edition, we have included additional material for use in modern applications of stochastic calculus in finance and biology; in particular, the section on infinitely divisible distributions and stable laws in Chap. 1, L´evy processes in Chap. 2, the Itˆ o–L´evy calculus in Chap. 3, and Chap. 4. Finally, a new appendix has been added that includes basic facts about semigroups of linear operators. We have also made an effort to improve the presentation of parts already included in the first edition, and we have corrected the misprints and errors we have been made aware of by colleagues and students during class use of the book in the intervening years. We are very grateful to all those who helped us in detecting them and suggested possible improvements. We are very grateful to Giacomo Aletti, Enea Bongiorno, Daniela Morale, Stefania Ugolini, and Elena Villa for checking the final proofs and suggesting valuable changes. Enea Bongiorno deserves special mention for his accurate editing of the book as you now see it. Tom Grasso from Birkh¨auser deserves acknowledgement for encouraging the preparation of a second, updated edition. Milan, Italy Milan, Italy Vincenzo Capasso David Bakstein vii Preface to the First Edition This book is a systematic, rigorous, and self-contained introduction to the theory of continuous-time stochastic processes. But it is neither a tract nor a recipe book as such; rather, it is an account of fundamental concepts as they appear in relevant modern applications and the literature. We make no pretense of its being complete. Indeed, we have omitted many results that we feel are not directly related to the main theme or that are available in easily accessible sources. Readers interested in the historical development of the subject cannot ignore the volume edited by Wax (1954). Proofs are often omitted as technicalities might distract the reader from a conceptual approach. They are produced whenever they might serve as a guide to the introduction of new concepts and methods to the applications; otherwise, explicit references to standard literature are provided. A mathematically oriented student may find it interesting to consider proofs as exercises. The scope of the book is profoundly educational, related to modeling realworld problems with stochastic methods. The reader becomes critically aware of the concepts involved in current applied literature and is, moreover, provided with a firm foundation of mathematical techniques. Intuition is always supported by mathematical rigor. Our book addresses three main groups of readers: first, mathematicians working in a different field; second, other scientists and professionals from a business or academic background; third, graduate or advanced undergraduate students of a quantitative subject related to stochastic theory or applications. As stochastic processes (compared to other branches of mathematics) are relatively new, yet increasingly popular in terms of current research output and applications, many pure as well as applied deterministic mathematicians have become interested in learning about the fundamentals of stochastic theory and modern applications. This book is written in a language that both groups will understand, and its content and structure will allow them to ix x Preface to the First Edition learn the essentials profoundly and in a time-efficient manner. Other scientistpractitioners and academics from fields like finance, biology, and medicine might be very familiar with a less mathematical approach to their specific fields and thus be interested in learning the mathematical techniques of modeling their applications. Furthermore, this book would be suitable as a textbook accompanying a graduate or advanced undergraduate course or as secondary reading for students of mathematical or computational sciences. The book has evolved from course material that has already been tested for many years in various courses in engineering, biomathematics, industrial mathematics, and mathematical finance. Last, but certainly not least, this book should also appeal to anyone who would like to learn about the mathematics of stochastic processes. The reader will see that previous exposure to probability, though helpful, is not essential and that the fundamentals of measure and integration are provided in a selfcontained way. Only familiarity with calculus and some analysis is required. The book is divided into three main parts. In Part I, comprising Chaps. 1–4, we introduce the foundations of the mathematical theory of stochastic processes and stochastic calculus, thereby providing the tools and methods needed in Part II (Chaps. 6 and 7), which is dedicated to major scientific areas of application. The third part consists of appendices, each of which gives a basic introduction to a particular field of fundamental mathematics (e.g., measure, integration, metric spaces) and explains certain problems in greater depth (e.g., stability of ODEs) than would be appropriate in the main part of the text. In Chap. 1 the fundamentals of probability are provided following a standard approach based on Lebesgue measure theory due to Kolmogorov. Here the guiding textbook on the subject is the excellent monograph by M´etivier (1968). Basic concepts from Lebesgue measure theory are also provided in Appendix A. Chapter 2 gives an introduction to the mathematical theory of stochastic processes in continuous time, including basic definitions and theorems on processes with independent increments, martingales, and Markov processes. The two fundamental classes of processes, Poisson and Wiener, are introduced as well as the larger, more general, class of L´evy processes. Further, a significant introduction to marked point processes is also given as a support for the analysis of relevant applications. Chapter 3 is based on Itˆo theory. We define the Itˆo integral, some fundamental results of Itˆo calculus, and stochastic differentials including Itˆo’s formula, as well as related results like the martingale representation theorem. Chapter 4 is devoted to the analysis of stochastic differential equations driven by Wiener processes and Itˆo diffusions and demonstrates the connections with partial differential equations of second order, via Dynkin and Feynman–Kac formulas. Preface to the First Edition xi Chapter 6 is dedicated to financial applications. It covers the core economic concept of arbitrage-free markets and shows the connection with martingales and Girsanov’s theorem. It explains the standard Black–Scholes theory and relates it to Kolmogorov’s partial differential equations and the Feynman–Kac formula. Furthermore, extensions and variations of the standard theory are discussed as are interest rate models and insurance mathematics. Chapter 7 presents fundamental models of population dynamics such as birth and death processes. Furthermore, it deals with an area of important modern research—the fundamentals of self-organizing systems, in particular focusing on the social behavior of multiagent systems, with some applications to economics (“price herding”). It also includes a particular application to the neurosciences, illustrating the importance of stochastic differential equations driven by both Poisson and Wiener processes. Problems and additions are proposed at the end of the volume, listed by chapter. In addition to exercises presented in a classical way, problems are proposed as a stimulus for discussing further concepts that might be of interest to the reader. Various sources have been used, including a selection of problems submitted to our students over the years. This is why we can provide only selected references. The core of this monograph, on Itˆo calculus, was developed during a series of courses that one of the authors, VC, has been offering at various levels in many universities. That author wishes to acknowledge that the first drafts of the relevant chapters were the outcome of a joint effort by many participating students: Maria Chiarolla, Luigi De Cesare, Marcello De Giosa, Lucia Maddalena, and Rosamaria Mininni, among others. Professor Antonio Fasano is due our thanks for his continuous support, including producing such material as lecture notes within a series that he coordinated. It was the success of these lecture notes, and the particular enthusiasm of coauthor DB, who produced the first English version (indeed, an unexpected Christmas gift), that has led to an extension of the material up to the present status, including, in particular, a set of relevant and updated applications that reflect the interests of the two authors. VC would also like to thank his first advisor and teacher, Professor Grace Yang, who gave him the first rigorous presentation of stochastic processes and mathematical statistics at the University of Maryland at College Park, always referring to real-world applications. DB would like to thank the Meregalli and Silvestri families for their kind logistical help while he was in Milan. He would also like to acknowledge research funding from the EPSRC, ESF, Socrates–Erasmus, and Charterhouse and thank all the people he worked with at OCIAM, University of Oxford, over the years, as this is where he was based when embarking on this project. The draft of the final volume was carefully read by Giacomo Aletti, Daniela Morale, Alessandra Micheletti, Matteo Ortisi, and Enea Bongiorno (who also took care of the problems and additions) whom we gratefully acknowledge. xii Preface to the First Edition Still, we are sure that some odd typos and other, hopefully noncrucial, mistakes remain, for which the authors take full responsibility. We also wish to thank Professor Nicola Bellomo, editor of the “Modeling and Simulation in Science, Engineering and Technology” series, and Tom Grasso from Birkh¨auser for supporting the project. Last but not least, we cannot neglect to thank Rossana (VC) and Casilda (DB) for their patience and great tolerance while coping with their “solitude” during the preparation of this monograph. Milan, Italy Milan, Italy Vincenzo Capasso David Bakstein Contents Preface to the Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Part I Theory of Stochastic Processes 1 Fundamentals of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Probability and Conditional Probability . . . . . . . . . . . . . . . . . . . . 1.2 Random Variables and Distributions . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Random Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Gaussian Random Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Conditional and Joint Distributions . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Convergence of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Infinitely Divisible Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Stable Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.1 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Exercises and Additions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 9 14 17 20 30 33 40 48 57 64 66 69 71 2 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Canonical Form of a Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 77 85 86 xiii xiv Contents 2.4 L2 Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.4.1 Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.4.2 Karhunen-Lo`eve Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.5 Processes with Independent Increments . . . . . . . . . . . . . . . . . . . . 91 2.6 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.7 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 2.7.1 The martingale problem for Markov processes . . . . . . . . . 124 2.8 Brownian Motion and the Wiener Process . . . . . . . . . . . . . . . . . . 129 2.9 Counting, and Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 145 2.10 Random Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 2.10.1 Poisson Random Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 151 2.11 Marked Counting Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 2.11.1 Counting Processes . . ...
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