Stochastic Integration and Differential Equations

It has been 15 years since the first edition of Stochastic Integration and Differential Equations , A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach". The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov criteria for exponential local martingales to be martingales, and a modern treatment of compensators. Chapter 4 treats sigma martingales (important in finance theory) and gives a more comprehensive treatment of martingale representation, including both the Jacod-Yor theory and Emery’s examples of martingales that actually have martingale representation (thus going beyond the standard cases of Brownian motion and the compensated Poisson process). New topics added include an introduction to the theory of the expansion of filtrations, a treatment of the Fefferman martingale inequality, and that the dual space of the martingale space H^1 can be identified with BMO martingales. Solutions to selected exercises are available at the web site of the author, with current URL http://www.orie.cornell.edu/~protter/books.html.

It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it a new approach.

The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov criteria for exponential local martingales to be martingales, and a modern treatment of compensators. Chapter 4 treats sigma martingales (important in finance theory) and gives a more comprehensive treatment of martingale representation, including both the Jacod-Yor theory and Emery’s examples of martingales that actually have martingale representation (thus going beyond the standard cases of Brownian motion and the compensated Poisson process). New topics added include an introduction to the theory of the expansion of filtrations, a treatment of the Fefferman martingale inequality, and that the dual space of the martingale space Hsub1 can be identified with BMO martingales. Solutions to selected exercises are available at the web site of the author, with current URL http://www.orie.cornell.edu/~protter/books.html.

Booknews

An introduction to the subject, requiring some knowledge of the theory of stochastic processes, including elementary martingale theory. The new approach considers semimartingales as good integrators rather than as the sum of a local martingale and a finite variation process. Examples such as Brownian motion, the Poisson process, and Levy processes are treated in detail. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Cover Title Preface to the Second Edition Preface to the First Edition Contents Introduction I Preliminaries 1 Basic Definitions and Notation 2 Martingales 3 The Poisson Process and Brownian Motion 4 Levy Processes 5 Why the Usual Hypotheses? 6 Local Martingales 7 Stieltjes Integration and Change of Variables 8 Naive Stochastic Integration Is Impossible Bibliographic Notes Exercises for Chapter I II Semimartingales and Stochastic Integrals 1 Introduction to Semimartingales 2 Stability Properties of Semimartingales 3 Elementary Examples of Semimartingales 4 Stochastic Integrals 5 Properties of Stochastic Integrals 6 The Quadratic Variation of a Semimartingale 7 Ito's Formula (Change of Variables) 8 Applications of Ito's Formula Bibliographic Notes Exercises for Chapter II III Semimartingales and Decomposable Processes 1 Introduction 2 The Classification of Stopping Times 3 The Doob-Meyer Decompositions 4 Quasimartingales 5 Compensators 6 The Fundamental Theorem of Local Martingales 7 Classical Semimartingales 8 Girsanov's Theorem 9 The Bichteler-Dellacherie Theorem Bibliographic Notes Exercises for Chapter III IV General Stochastic Integration and Local Times 1 Introduction 2 Stochastic Integration for Predictable Integrands 3 Martingale Representation 4 Martingale Duality and the Jacod-Yor Theorem on Martingale Representation 5 Examples of Martingale Representation 6 Stochastic Integration Depending on a Parameter 7 Local Times 8 Azema's Martingale 9 Sigma Martingales Bibliographic Notes Exercises for Chapter IV V Stochastic Differential Equations 1 Introduction 2 The È? Norms for Semimartingales 3 Existence and Uniqueness of Solutions 4 Stability of Stochastic Differential Equations 5 Fisk-Stratonovich Integrals and Differential Equations 6 The Markov Nature of Solutions 7 Flows of Stochastic Differential Equations: Continuity and Differentiability 8 Flows as Diffeomorphisms: The Continuous Case 9 General Stochastic Exponentials and Linear Equations 10 Flows as Diffeomorphisms: The General Case 11 Eclectic Useful Results on Stochastic Differential Equations . .. Bibliographic Notes Exercises for Chapter V VI Expansion of Filtrations 1 Introduction 2 Initial Expansions 3 Progressive Expansions 4 Time Reversal Bibliographic Notes Exercises for Chapter VI References Symbol Index Subject Index It has been 13 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though we will no longer call it a new approach . The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises! Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been nearly completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov criteria for exponential local martingales to be martingales, and a modern treatment of compensators. Chapter 4 treats sigma martingales (important in finance theory) and gives a more comprehensive treatment of martingale representation, including both the Jacod-Yor theory and Emery's examples of martingales that actually have martingale representation (thus going beyond the standard cases of Brownian motion and the compensated Poisson process). New topics added include an introduction to the theory of the expansion of filtrations, and an elementary treatment of the Burkholder-Gundy-Fefferman martingale inequalities. Last, there are of course small changes throughout the book