**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
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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
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© 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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