随机过程：Introduction to Malliavin calculus by Nualart, David Nualart, Eulalia

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Contents
Preface page xi
1 Brownian Motion 1
1.1 Preliminaries and Notation 1
1.2 Definition and Basic Properties 1
1.3 Wiener Integral 7
1.4 Wiener Space 9
1.5 Brownian Filtration 9
1.6 Markov Property 10
1.7 Martingales Associated with Brownian Motion 11
1.8 Strong Markov Property 14
Exercises 16
2 Stochastic Calculus 18
2.1 Stochastic Integrals 18
2.2 Indefinite Stochastic Integrals 23
2.3 Integral of General Processes 28
2.4 Itô’s Formula 30
2.5 Tanaka’s Formula 35
2.6 Multidimensional Version of Itô’s Formula 38
2.7 Stratonovich Integral 40
2.8 Backward Stochastic Integral 41
2.9 Integral Representation Theorem 42
2.10 Girsanov’s Theorem 44
Exercises 47
3 Derivative and Divergence Operators 50
3.1 Finite-Dimensional Case 50
3.2 Malliavin Derivative 51
3.3 Sobolev Spaces 53
3.4 The Divergence as a Stochastic Integral 56
vii
viii Contents
3.5 Isonormal Gaussian Processes 57
Exercises 61
4 Wiener Chaos 63
4.1 Multiple Stochastic Integrals 63
4.2 Derivative Operator on the Wiener Chaos 65
4.3 Divergence on the Wiener Chaos 68
4.4 Directional Derivative 69
Exercises 72
5 Ornstein–Uhlenbeck Semigroup 74
5.1 Mehler’s Formula 74
5.2 Generator of the Ornstein–Uhlenbeck Semigroup 78
5.3 Meyer’s Inequality 80
5.4 Integration-by-Parts Formula 83
5.5 Nourdin–Viens Density Formula 84
Exercises 86
6 Stochastic Integral Representations 87
6.1 Clark–Ocone formula 87
6.2 Modulus of Continuity of the Local Time 90
6.3 Derivative of the Self-Intersection Local Time 96
6.4 Application of the Clark–Ocone Formula in Finance 97
6.5 Second Integral Representation 99
6.6 Proving Tightness Using Malliavin Calculus 100
Exercises 103
7 Study of Densities 105
7.1 Analysis of Densities in the One-Dimensional Case 105
7.2 Existence and Smoothness of Densities for Random Vectors 108
7.3 Density Formula using the Riesz Transform 111
7.4 Log-Likelihood Density Formula 113
7.5 Malliavin Differentiability of Diffusion Processes 118
7.6 Absolute Continuity under Ellipticity Conditions 122
7.7 Regularity of the Density under Hörmander’s Conditions 123
Exercises 129
8 Normal Approximations 131
8.1 Stein’s Method 131
8.2 Stein Meets Malliavin 136
8.3 Normal Approximation on a Fixed Wiener Chaos 138
8.4 Chaotic Central Limit Theorem 143
8.5 Applications to Fractional Brownian Motion 146
Contents ix
8.6 Convergence of Densities 150
8.7 Noncentral Limit Theorems 153
Exercises 156
9 Jump Processes 158
9.1 Lévy Processes 158
9.2 Poisson Random Measures 160
9.3 Integral with respect to a Poisson Random Measure 163
9.4 Stochastic Integrals with respect to the Jump Measure of a
Lévy Process 164
9.5 Itô’s Formula 168
9.6 Integral Representation Theorem 172
9.7 Girsanov’s Theorem 174
9.8 Multiple Stochastic Integrals 175
9.9 Wiener Chaos for Poisson Random Measures 177
Exercises 180
10 Malliavin Calculus for Jump Processes I 182
10.1 Derivative Operator 182
10.2 Divergence Operator 187
10.3 Ornstein–Uhlenbeck Semigroup 191
10.4 Clark–Ocone Formula 192
10.5 Stein’s Method for Poisson Functionals 193
10.6 Normal Approximation on a Fixed Chaos 194
Exercises 199
11 Malliavin Calculus for Jump Processes II 201
11.1 Derivative Operator 201
11.2 Sobolev Spaces 205
11.3 Directional Derivative 208
11.4 Application to Diffusions with Jumps 212
Exercises 220
Appendix A Basics of Stochastic Processes 221
A.1 Stochastic Processes 221
A.2 Gaussian Processes 222
A.3 Equivalent Processes 223
A.4 Regularity of Trajectories 223
A.5 Markov Processes 223
A.6 Stopping Times 224
A.7 Martingales 225
References 228
Index 235