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Introduction 1

1 Analysis on the Wiener space 3

1.1 Wiener chaos and stochastic integrals 3

1.1.1 The Wiener chaos decomposition 4

1.1.2 The white noise case:Multiple Wiener-It？ integrals 8

1.1.3 It？ stochastic calculus 15

1.2 The derivative operator 24

1.2.1 The derivative operator in the white noise case 31

1.3 The divergence operator 36

1.3.1 Properties of the divergence operator 37

1.3.2 The Skorohod integral 40

1.3.3 The It？ stochastic integral as a particular case of the Skorohod integral 44

1.3.4 Stochastic integral representation of Wiener functionals 46

1.3.5 Local properties 47

1.4 The Ornstein-Uhlenbeck semigroup 54

1.4.1 The semigroup of Ornstein-Uhlenbeck 54

1.4.2 The generator of the Ornstein-Uhlenbeck semigroup 58

1.4.3 Hypercontractivity property and the multiplier theorem 61

1.5 Sobolev spaces and the equivalence of norms 67

2 Regularity of probability laws 85

2.1 Regularity of densities and related topics 85

2.1.1 Computation and estimation of probability densities 86

2.1.2 A criterion for absolute continuity based on the integration-by-parts formula 90

2.1.3 Absolute continuity using Bouleau and Hirsch's ap-proach 94

2.1.4 Smoothness of densities 99

2.1.5 Composition of tempered distributions with nonde-generate random vectors 104

2.1.6 Properties of the support of the law 105

2.1.7 Regularity of the law of the maximum of continuous processes 108

2.2 Stochastic differential equations 116

2.2.1 Existence and uniqueness of solutions 117

2.2.2 Weak differentiability of the solution 119

2.3 Hypoellipticity and H？rmander's theorem 125

2.3.1 Absolute continuity in the case of Lipschitz coefficients 125

2.3.2 Absolute continuity under H？rmander's conditions 128

2.3.3 Smoothness of the density under H？rmander's condition 133

2.4 Stochastic partial differential equations 142

2.4.1 Stochastic integral equations on the plane 142

2.4.2 Absolute continuity for solutions to the stochastic heat equation 151

3 Anticipating stochastic calculus 169

3.1 Approximation of stochastic integrals 169

3.1.1 Stochastic integrals defined by Riemann sums 170

3.1.2 The approach based on the L2 development of the process 176

3.2 Stochastic calculus for anticipating integrals 180

3.2.1 Skorohod integral processes 180

3.2.2 Continuity and quadratic variation of the Skorohod integral 181

3.2.3 It？'s formula for the Skorohod and Stratonovich integrals 184

3.2.4 Substitution formulas 195

3.3 Anticipating stochastic differential equations 208

3.3.1 Stochastic differential equations in the Sratonovich sense 208

3.3.2 Stochastic differential equations with boundary con-ditions 215

3.3.3 Stochastic differential equations in the Skorohod sense 217

4 Transformations of the Wiener measure 225

4.1 Anticipating Girsanov theorems 225

4.1.1 The adapted case 226

4.1.2 General results on absolute continuity of transformations 228

4.1.3 Continuously differentiable variables in the direction of H1 230

4.1.4 Transformations induced by elementary processes 232

4.1.5 Anticipating Girsanov theorems 234

4.2 Markov random fields 241

4.2.1 Markov field property for stochastic differential equations with boundary conditions 242

4.2.2 Markov field property for solutions to stochastic partial differential equations 249

4.2.3 Conditional independence and factorization properties 258

5 Fractional Brownian motion 273

5.1 Definition,properties and construction of the fractional Brown-ian motion 273

5.1.1 Semimartingale property 274

5.1.2 Moving average representation 276

5.1.3 Representation of fBm on an interval 277

5.2 Stochastic calculus with respect to fBm 287

5.2.1 Malliavin Calculus with respect to the fBm 287

5.2.2 Stochastic calculus with respect to fBm.Case H＞1/2 288

5.2.3 Stochastic integration with respect to fBm in the case H＜1/2 295

5.3 Stochastic differential equations driven by a fBm 306

5.3.1 Generalized Stieltjes integrals 306

5.3.2 Deterministic differential equations 309

5.3.3 Stochastic differential equations with respect to fBm 312

5.4 Vortex filaments based on fBm 313

6 Malliavin Calculus in finance 321

6.1 Black-Scholes model 321

6.1.1 Arbitrage opportunities and martingale measures 323

6.1.2 Completeness and hedging 325

6.1.3 Black-Scholes formula 327

6.2 Integration by parts formulas and computation of Greeks 330

6.2.1 Computation of Greeks for European options 332

6.2.2 Computation of Greeks for exotic options 334

6.3 Application of the Clark-Ocone formula in hedging 336

6.3.1 A generalized Clark-Ocone formula 336

6.3.2 Application to finance 338

6.4 Insider trading 340

A Appendix 351

A.1 A Gaussian formula 351

A.2 Martingale inequalities 351

A.3 Continuity criteria 353

A.4 Carleman-Fredholm determinant 354

A.5 Fractional integrals and derivatives 355

References 357

Index 377