The Malliavin Calculus and Related Topics

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-Itˆo integrals . 8

1.1.3 Itˆo 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 Itˆo 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

xii Contents

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 approach

. . . . . . . . . . . . . . . . . . . . . . . . . . 94

2.1.4 Smoothness of densities . . . . . . . . . . . . . . . . 99

2.1.5 Composition of tempered distributions with nondegenerate

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¨ormander’s theorem . . . . . . . . . . 125

2.3.1 Absolute continuity in the case

of Lipschitz coefficients . . . . . . . . . . . . . . . . 125

2.3.2 Absolute continuity under H¨ormander’s conditions . 128

2.3.3 Smoothness of the density

under H¨ormander’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 Itˆo’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 conditions

. . . . . . . . . . . . . . . . . . . . . . . . . . 215

Contents xiii

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 Brownian

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 caseH <

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

xiv Contents

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