Stochastic Integration and Differential Equations

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Springer Science & Business Media, 4 mars 2005 - 415 pages
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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.

 

 

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Table des matières

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
References
397
Subject Index
411
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