Stochastic Integration and Differential EquationsIt 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 DoobMeyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the KazamakiNovikov 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 JacodYor 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.

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Contents
Introduction  1 
Preliminaries  3 
2 Martingales  7 
3 The Poisson Process and Brownian Motion  12 
4 Lévy Processes  20 
5 Why the Usual Hypotheses?  34 
6 Local Martingales  37 
7 Stieltjes Integration and Change of Variables  39 
General Stochastic Integration and Local Times  155 
3 Martingale Representation  180 
4 Martingale Duality and the JacodYor Theorem on Martingale Representation  195 
5 Examples of Martingale Representation  203 
6 Stochastic Integration Depending on a Parameter  208 
7 Local Times  213 
8 Azémas Martingale  232 
9 Sigma Martingales  237 
8 Naïve Stochastic Integration is Impossible  43 
Bibliographic Notes  44 
Exercises for Chapter I  45 
Semimartingales and Stochastic Integrals  51 
2 Stability Properties of Semimartingales  52 
3 Elementary Examples of Semimartingales  54 
4 Stochastic Integrals  56 
5 Properties of Stochastic Integrals  60 
6 The Quadratic Variation of a Semimartingale  66 
7 Itôs Formula Change of Variables  78 
8 Applications of Itôs Formula  84 
Bibliographic Notes  92 
Exercises for Chapter II  94 
Semimartingales and Decomposable Processes  101 
2 The Classification of Stopping Times  104 
3 The DoobMeyer Decompositions  106 
4 Quasimartingales  117 
5 Compensators  119 
6 The Fundamental Theorem of Local Martingales  126 
7 Classical Semimartingales  129 
8 Girsanovs Theorem  133 
9 The BichtelerDellacherie Theorem  146 
Bibliographic Notes  149 
Exercises for Chapter III  150 
Bibliographic Notes  240 
Exercises for Chapter IV  241 
Stochastic Differential Equations  249 
2 The Hᴾ Norms for Semimartingales  250 
3 Existence and Uniqueness of Solutions  255 
4 Stability of Stochastic Differential Equations  263 
5 FiskStratonovich Integrals and Differential Equations  277 
6 The Markov Nature of Solutions  297 
Continuity and Differentiability  307 
The Continuous Case  317 
9 General Stochastic Exponentials and Linear Equations  328 
The General Case  335 
11 Eclectic Useful Results on Stochastic Differential Equations  345 
Bibliographic Notes  354 
Exercises for Chapter V  355 
Expansion of Filtrations  363 
2 Initial Expansions  364 
3 Progressive Expansions  378 
4 Time Reversal  385 
Bibliographic Notes  391 
Exercises for Chapter VI  392 
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