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目录：

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 Basic Definitions and Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 The Poisson Process and Brownian Motion ................. 12
4 Levy Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19
5 Why the Usual Hypotheses? .............................. 34
6 Local Martingales ....................................... 37
7 Stieltjes Integration and Change of Variables. . . . . . . . . . . . . . .. 39
8 NaIve Stochastic Integration Is Impossible. . . . . . . . . . . . . . . . .. 43
Bibliographic Notes .................................... . . . . .. 44
Exercises for Chapter I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45

II Semimartingales and Stochastic Integrals .................. 51
1 Introduction to Semimartingales. . . . . . . . . . . . . . . . . . . . . . . . . .. 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 Ito's Formula (Change of Variables). . . . . . . . . . . . . . . . . . . . . . .. 78
8 Applications of Ito's Formula ............................. 84
Bibliographic Notes ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92
Exercises for Chapter II ...................................... 94

III Semimartingales and Decomposable Processes ............. 101
1 Introduction ............................................ 101
2 The Classification of Stopping Times ....................... 103
3 The Doob-Meyer Decompositions .......................... 105
4 Quasimartingales ........................................ 116
5 Compensators ........................................... 118
6 The Fundamental Theorem of Local Martingales ............ 124
7 Classical Semimartingales ................................ 127
8 Girsanov's Theorem ..................................... 131
9 The Bichteler-Dellacherie Theorem ........................ 143
Bibliographic Notes .......................................... 147
Exercises for Chapter III ...................................... 147

IV General Stochastic Integration and Local Times ........... 153
1 Introduction ............................................ 153
2 Stochastic Integration for Predictable Integrands ............ 153
3 Martingale Representation ................................ 178
4 Martingale Duality and the Jacod-Yor Theorem on
Martingale Representation ................................ 193
5 Examples of Martingale Representation .................... 200
6 Stochastic Integration Depending on a Parameter ............ 205
7 Local Times ............................................ 210
8 Azema's Martingale ...................................... 227
9 Sigma Martingales ....................................... 233
Bibliographic Notes .......................................... 235
Exercises for Chapter IV ...................................... 236

V Stochastic Differential Equations .......................... 243
1 Introduction ............................................ 243
2 The HP Norms for Semimartingales ........................ 244
3 Existence and Uniqueness of Solutions ..................... 249
4 Stability of Stochastic Differential Equations ................ 257
5 Fisk-Stratonovich Integrals and Differential Equations ........ 270
6 The Markov Nature of Solutions ........................... 291
7 Flows of Stochastic Differential Equations: Continuity and
Differentiability ......................................... 301
8 Flows as Diffeomorphisms: The Continuous Case ............ 310
9 General Stochastic Exponentials and Linear Equations ....... 321
10 Flows as Diffeomorphisms: The General Case ............... 328
11 Eclectic Useful Results on Stochastic Differential Equations ... 338
Bibliographic Notes .......................................... 347
Exercises for Chapter V ...................................... 349

VI Expansion of Filtrations ................................... 355
1 Introduction ............................................ 355
2 Initial Expansions ....................................... 356
3 Progressive Expansions ................................... 369
4 Time Reversal .......................................... 377
Bibliographic Notes .......................................... 383
Exercises for Chapter VI ...................................... 384