Malliavi随机分析和相关论题 第2版 英文PDF|Epub|txt|kindle电子书版本网盘下载

Malliavi随机分析和相关论题 第2版 英文PDF格式电子书版下载

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Introduction1

1 Analysis on the Wiener space3

1.1 Wiener chaos and stochastic integrals3

1.1.1 The Wiener chaos decomposition4

1.1.2 The white noise case:Multiple Wiener-It？ integrals8

1.1.3 It？ stochastic calculus15

1.2 The derivative operator24

1.2.1 The derivative operator in the white noise case31

1.3 The divergence operator36

1.3.1 Properties of the divergence operator37

1.3.2 The Skorohod integral40

1.3.3 The It？ stochastic integral as a particular case of the Skorohod integral44

1.3.4 Stochastic integral representation of Wiener functionals46

1.3.5 Local properties47

1.4 The Ornstein-Uhlenbeck semigroup54

1.4.1 The semigroup of Ornstein-Uhlenbeck54

1.4.2 The generator of the Ornstein-Uhlenbeck semigroup58

1.4.3 Hypercontractivity property and the multiplier theorem61

1.5 Sobolev spaces and the equivalence of norms67

2 Regularity of probability laws85

2.1 Regularity of densities and related topics85

2.1.1 Computation and estimation of probability densities86

2.1.2 A criterion for absolute continuity based on the integration-by-parts formula90

2.1.3 Absolute continuity using Bouleau and Hirsch's ap-proach94

2.1.4 Smoothness of densities99

2.1.5 Composition of tempered distributions with nonde-generate random vectors104

2.1.6 Properties of the support of the law105

2.1.7 Regularity of the law of the maximum of continuous processes108

2.2 Stochastic differential equations116

2.2.1 Existence and uniqueness of solutions117

2.2.2 Weak differentiability of the solution119

2.3 Hypoellipticity and H？rmander's theorem125

2.3.1 Absolute continuity in the case of Lipschitz coefficients125

2.3.2 Absolute continuity under H？rmander's conditions128

2.3.3 Smoothness of the density under H？rmander's condition133

2.4 Stochastic partial differential equations142

2.4.1 Stochastic integral equations on the plane142

2.4.2 Absolute continuity for solutions to the stochastic heat equation151

3 Anticipating stochastic calculus169

3.1 Approximation of stochastic integrals169

3.1.1 Stochastic integrals defined by Riemann sums170

3.1.2 The approach based on the L2 development of the process176

3.2 Stochastic calculus for anticipating integrals180

3.2.1 Skorohod integral processes180

3.2.2 Continuity and quadratic variation of the Skorohod integral181

3.2.3 It？'s formula for the Skorohod and Stratonovich integrals184

3.2.4 Substitution formulas195

3.3 Anticipating stochastic differential equations208

3.3.1 Stochastic differential equations in the Sratonovich sense208

3.3.2 Stochastic differential equations with boundary con-ditions215

3.3.3 Stochastic differential equations in the Skorohod sense217

4 Transformations of the Wiener measure225

4.1 Anticipating Girsanov theorems225

4.1.1 The adapted case226

4.1.2 General results on absolute continuity of transformations228

4.1.3 Continuously differentiable variables in the direction of H1230

4.1.4 Transformations induced by elementary processes232

4.1.5 Anticipating Girsanov theorems234

4.2 Markov random fields241

4.2.1 Markov field property for stochastic differential equations with boundary conditions242

4.2.2 Markov field property for solutions to stochastic partial differential equations249

4.2.3 Conditional independence and factorization properties258

5 Fractional Brownian motion273

5.1 Definition,properties and construction of the fractional Brown-ian motion273

5.1.1 Semimartingale property274

5.1.2 Moving average representation276

5.1.3 Representation of fBm on an interval277

5.2 Stochastic calculus with respect to fBm287

5.2.1 Malliavin Calculus with respect to the fBm287

5.2.2 Stochastic calculus with respect to fBm.Case H＞1/2288

5.2.3 Stochastic integration with respect to fBm in the case H＜1/2295

5.3 Stochastic differential equations driven by a fBm306

5.3.1 Generalized Stieltjes integrals306

5.3.2 Deterministic differential equations309

5.3.3 Stochastic differential equations with respect to fBm312

5.4 Vortex filaments based on fBm313

6 Malliavin Calculus in finance321

6.1 Black-Scholes model321

6.1.1 Arbitrage opportunities and martingale measures323

6.1.2 Completeness and hedging325

6.1.3 Black-Scholes formula327

6.2 Integration by parts formulas and computation of Greeks330

6.2.1 Computation of Greeks for European options332

6.2.2 Computation of Greeks for exotic options334

6.3 Application of the Clark-Ocone formula in hedging336

6.3.1 A generalized Clark-Ocone formula336

6.3.2 Application to finance338

6.4 Insider trading340

A Appendix351

A.1 A Gaussian formula351

A.2 Martingale inequalities351

A.3 Continuity criteria353

A.4 Carleman-Fredholm determinant354

A.5 Fractional integrals and derivatives355

References357

Index377