1 Introduction to Stochastic Processes ......................... l
1.1 The Kolmogorov Consistency Theorem ...................... 1
1.2 The Language of Stochastic Processes ................... 11
1.3 Sigma Fields, Measurability, and Stopping Times ........ 14
Exercises ................................................... 17
2 Brownian Motion ............................................. 19
2.1 Definition and Construction of Brownian Motion ......... 20
2.2 Essential Features of a Brownian Motion ................ 27
2.3 The Reflection Principle ............................... 34
Exercises ................................................... 39
3 Elements of Martingale Theory ............................... 41
3.1 Definition and Examples of Martingales ................. 41
3.2 Wiener Martingales and the Markov Property ............. 44
3.3 Essential Results on Martingales ....................... 49
3.4 The Doob-Meyer Decomposition ........................... 54
3.5 The Meyer Process for L2-martingales ................... 67
3.6 Local Martingales ...................................... 71
Exercises ................................................... 73
4 Analytical Tools for Brownian Motion ........................ 75
4.1 Introduction ........................................... 75
4.2 The Brownian Semigroup ................................. 76
4.3 Resolvents and Generators .............................. 79
4.4 Pregenerators and Martingales .......................... 87
Exercises ................................................... 89
5 Stochastic Integration ...................................... 90
5.1 The Itô Integral ....................................... 90
5.2 Properties of the Integral ............................. 98
5.3 Vector-valued Processes ............................... 105
5.4 The Itô Formula ....................................... 106
5.5 An Extension of the Itô Formula ....................... 111
5.6 Applications of the Itô Formula ....................... 113
5.7 The Girsanov Theorem .................................. 124
Exercises .................................................. 132
6 Stochastic Differential Equations .......................... 134
6.1 Introduction .......................................... 134
6.2 Existence and Uniqueness of Solutions ................. 137
6.3 Linear Stochastic Differential Equations .............. 144
6.4 Weak Solutions ........................................ 146
6.5 Markov Property ....................................... 153
6.6 Generators and Diffusion Processes .................... 161
Exercises .................................................. 164
7 The Martingale Problem ..................................... 166
7.1 Introduction .......................................... 166
7.2 Existence of Solutions ................................ 174
7.3 Analytical Tools ...................................... 183
7.4 Uniqueness of Solutions ............................... 189
7.5 Markov Property of Solutions .......................... 193
7.6 Further Results on Uniqueness ......................... 196
8 Probability Theory and Partial Differential Equations ...... 202
8.1 The Dirichlet Problem ................................. 202
8.2 Boundary Regularity ................................... 212
8.3 Kolmogorov Equations: The Heuristics .................. 218
8.4 Feynman-Kac Formula ................................... 221
8.5 An Application to Finance Theory ...................... 223
8.6 Kolmogorov Equations .................................. 224
Exercises .................................................. 239
9 Gaussian Solutions ......................................... 240
9.1 Introduction .......................................... 241
9.2 Hilbert-Schmidt Operators ............................. 245
9.3 The Gohberg-Krein Factorization ....................... 248
9.4 Nonanticipative Representations ....................... 252
9.5 Gaussian Solutions of Stochastic Equations ............ 257
Exercises .................................................. 265
10 Jump Markov Processes ...................................... 266
10.1 Definitions and Basic Results ......................... 266
10.2 Stochastic Calculus for Processes with Jumps .......... 271
10.3 Jump Markov Processes ................................. 275
10.4 Diffusion Approximation ............................... 283
Exercises .................................................. 290
11 Invariant Measures and Ergodicity .......................... 292
11.1 Introduction .......................................... 293
11.2 Ergodicity for One-dimensional Diffusions ............. 295
11.3 Invariant Measures for d-dimensional Diffusions ....... 301
11.4 Existence and Uniqueness of Invariant Measures ........ 304
11.5 Ergodic Measures ...................................... 310
Exercises .................................................. 314
12 Large Deviations Principle for Diffusions .................. 315
12.1 Definitions and Basic Results ......................... 316
12.2 Large Deviations and Laplace-Varadhan Principle ....... 318
12.3 A Variational Representation Theorem .................. 329
12.4 Sufficient Conditions for LDP ......................... 338
Exercises .................................................. 341
Notes on Chapters ............................................. 343
References .................................................... 347
Index ......................................................... 351
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