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
