Preface ........................................................ ix
Frequently Used Notation ...................................... xii
1 Fundamentals of Continuous Stochastic Processes .............. 1
1.1 Stochastic Processes .................................... 1
1.2 Wiener Space ............................................ 4
1.3 Filtered Probability Space, Adapted Stochastic
Process ................................................. 9
1.4 Discrete Time Martingales .............................. 11
1.4.1 Conditional Expectation ......................... 11
1.4.2 Martingales, Doob Decomposition ................. 13
1.4.3 Optional Stopping Theorem ....................... 16
1.4.4 Convergence Theorem ............................. 17
1.4.5 Optional Sampling Theorem ....................... 20
1.4.6 Doob's Inequality ............................... 22
1.5 Continuous Time Martingale ............................. 24
1.5.1 Fundamentals .................................... 24
1.5.2 Examples on the Wiener Space .................... 25
1.5.3 Optional Sampling Theorem, Doob's Inequality,
Convergence Theorem ............................. 28
1.5.4 Applications .................................... 32
1.5.5 Doob-Meyer Decomposition, Quadratic Variation
Process ......................................... 34
1.6 Adapted Brownian Motion ................................ 37
1.7 Cameron-Martin Theorem ................................. 40
1.8 Schilder's Theorem ..................................... 43
1.9 Analogy to Path Integrals .............................. 49
2 Stochastic Integrals and Itô's Formula ...................... 52
2.1 Local Martingale ....................................... 52
2.2 Stochastic Integrals ................................... 54
2.3 Itô's Formula .......................................... 61
2.4 Moment Inequalities for Martingales .................... 70
2.5 Martingale Characterization of Brownian Motion ......... 73
2.6 Martingales with respect to Brownian Motions ........... 82
2.7 Local Time, Itô-Tanaka Formula ......................... 87
2.8 Reflecting Brownian Motion and Skorohod Equation ....... 93
2.9 Conformal Martingales .................................. 96
3 Brownian Motion and the Laplacian .......................... 102
3.1 Markov and Strong Markov Properties ................... 102
3.2 Recurrence and Transience of Brownian Motions ......... 108
3.3 Heat Equations ........................................ 111
3.4 Non-Homogeneous Equation .............................. 112
3.5 The Feynman-Kac Formula ............................... 117
3.6 The Dirichlet Problem ................................. 125
4 Stochastic Differential Equations .......................... 133
4.1 Introduction: Diffusion Processes ..................... 133
4.2 Stochastic Differential Equations ..................... 138
4.3 Existence of Solutions ................................ 145
4.4 Pathwise Uniqueness ................................... 151
4.5 Martingale Problems ................................... 156
4.6 Exponential Martingales and Transformation of Drift ... 157
4.7 Solutions by Time Change .............................. 164
4.8 One-Dimensional Diffusion Process ..................... 167
4.9 Linear Stochastic Differential Equations .............. 180
4.10 Stochastic Flows ...................................... 183
4.11 Approximation Theorem ................................. 190
5 Malliavin Calculus ......................................... 195
5.1 Sobolev Spaces and Differential Operators ............. 195
5.2 Continuity of Operators ............................... 206
5.3 Characterization of Sobolev Spaces .................... 214
5.4 Integration by Parts Formula .......................... 224
5.5 Application to Stochastic Differential Equations ...... 232
5.6 Change of Variables Formula ........................... 244
5.7 Quadratic Forms ....................................... 257
5.8 Examples of Quadratic Forms ........................... 265
5.8.1 Harmonic Oscillators ........................... 265
5.8.2 Lévy's Stochastic Area ......................... 269
5.8.3 Sample Variance ................................ 274
5.9 Abstract Wiener Spaces and Rough Paths ................ 276
6 The Black-Scholes Model .................................... 281
6.1 The Black-Scholes Model ............................... 281
6.2 Arbitrage Opportunity, Equivalent Martingale
Measures .............................................. 284
6.3 Pricing Formula ....................................... 287
6.4 Greeks ................................................ 293
7 The Semiclassical Limit .................................... 297
7.1 Van Vleck's Result and Quadratic Functionals .......... 297
7.1.1 Soliton Solutions for the KdV Equation ......... 302
7.1.2 Euler Polynomials .............................. 307
7.2 Asymptotic Distribution of Eigenvalues ................ 309
7.3 Semiclassical Approximation of Eigenvalues ............ 312
7.4 Selberg's Trace Formula on the Upper Half Plane ....... 318
7.5 Integral of Geometric Brownian Motion and Heat
Kernel on H2 .......................................... 323
Appendix. Some Fundamentals ................................... 329
References .................................................... 337
Index ......................................................... 344
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