The Malliavin calculus is an infinite-dimensional differential calculus on a Gaussian space, developed to provide a probabilistic proof to Hörmander's sum of squares theorem but has found a range of applications in stochastic analysis. This book presents the features of Malliavin calculus and discusses its main applications. This second edition includes recent applications in finance and a chapter devoted to the stochastic calculus with respect to the fractional Brownian motion.
Author(s): David Nualart
Series: Probability and Its Applications
Edition: 2nd
Publisher: Springer
Year: 2006
Language: English
Pages: 390
Contents......Page 11
Introduction......Page 15
1.1 Wiener chaos and stochastic integrals......Page 17
1.1.1 The Wiener chaos decomposition......Page 18
1.1.2 The white noise case: Multiple Wiener-Itô integrals......Page 22
1.1.3 Itô stochastic calculus......Page 29
1.2 The derivative operator......Page 38
1.2.1 The derivative operator in the white noise case......Page 45
1.3 The divergence operator......Page 50
1.3.1 Properties of the divergence operator......Page 51
1.3.2 The Skorohod integral......Page 54
1.3.3 The Itô stochastic integral as a particular case of the Skorohod integral......Page 58
1.3.4 Stochastic integral representation of Wiener functionals......Page 60
1.3.5 Local properties......Page 61
1.4.1 The semigroup of Ornstein-Uhlenbeck......Page 68
1.4.2 The generator of the Ornstein-Uhlenbeck semigroup......Page 72
1.4.3 Hypercontractivity property and the multiplier theorem......Page 75
1.5 Sobolev spaces and the equivalence of norms......Page 81
2.1 Regularity of densities and related topics......Page 98
2.1.1 Computation and estimation of probability densities......Page 99
2.1.2 A criterion for absolute continuity based on the integration-by-parts formula......Page 103
2.1.3 Absolute continuity using Bouleau and Hirsch’s approach......Page 107
2.1.4 Smoothness of densities......Page 112
2.1.5 Composition of tempered distributions with nondegenerate random vectors......Page 117
2.1.6 Properties of the support of the law......Page 118
2.1.7 Regularity of the law of the maximum of continuous processes......Page 121
2.2 Stochastic differential equations......Page 129
2.2.1 Existence and uniqueness of solutions......Page 130
2.2.2 Weak differentiability of the solution......Page 132
2.3.1 Absolute continuity in the case of Lipschitz coefficients......Page 138
2.3.2 Absolute continuity under Hörmander’s conditions......Page 141
2.3.3 Smoothness of the density under Hörmander’s condition......Page 146
2.4.1 Stochastic integral equations on the plane......Page 155
2.4.2 Absolute continuity for solutions to the stochastic heat equation......Page 164
3.1 Approximation of stochastic integrals......Page 181
3.1.1 Stochastic integrals defined by Riemann sums......Page 182
3.1.2 The approach based on the L[sup(2)] development of the process......Page 188
3.2.1 Skorohod integral processes......Page 192
3.2.2 Continuity and quadratic variation of the Skorohod integral......Page 193
3.2.3 Itô’s formula for the Skorohod and Stratonovich integrals......Page 196
3.2.4 Substitution formulas......Page 207
3.3.1 Stochastic differential equations in the Sratonovich sense......Page 220
3.3.2 Stochastic differential equations with boundary conditions......Page 227
3.3.3 Stochastic differential equations in the Skorohod sense......Page 229
4.1 Anticipating Girsanov theorems......Page 236
4.1.1 The adapted case......Page 237
4.1.2 General results on absolute continuity of transformations......Page 239
4.1.3 Continuously differentiable variables in the direction of H[sup(1)]......Page 241
4.1.4 Transformations induced by elementary processes......Page 243
4.1.5 Anticipating Girsanov theorems......Page 245
4.2 Markov random fields......Page 252
4.2.1 Markov field property for stochastic differential equations with boundary conditions......Page 253
4.2.2 Markov field property for solutions to stochastic partial differential equations......Page 260
4.2.3 Conditional independence and factorization properties......Page 269
5.1 Definition, properties and construction of the fractional Brownian motion......Page 283
5.1.1 Semimartingale property......Page 284
5.1.2 Moving average representation......Page 286
5.1.3 Representation of fBm on an interval......Page 287
5.2.1 Malliavin Calculus with respect to the fBm......Page 297
5.2.2 Stochastic calculus with respect to fBm. Case H > ½......Page 298
5.2.3 Stochastic integration with respect to fBm in the case H < ½......Page 305
5.3.1 Generalized Stieltjes integrals......Page 316
5.3.2 Deterministic differential equations......Page 319
5.3.3 Stochastic differential equations with respect to fBm......Page 322
5.4 Vortex filaments based on fBm......Page 323
6.1 Black-Scholes model......Page 331
6.1.1 Arbitrage opportunities and martingale measures......Page 333
6.1.2 Completeness and hedging......Page 335
6.1.3 Black-Scholes formula......Page 337
6.2 Integration by parts formulas and computation of Greeks......Page 340
6.2.1 Computation of Greeks for European options......Page 342
6.2.2 Computation of Greeks for exotic options......Page 344
6.3.1 A generalized Clark-Ocone formula......Page 346
6.3.2 Application to finance......Page 348
6.4 Insider trading......Page 350
A.2 Martingale inequalities......Page 360
A.3 Continuity criteria......Page 362
A.4 Carleman-Fredholm determinant......Page 363
A.5 Fractional integrals and derivatives......Page 364
References......Page 366
C......Page 385
K......Page 386
S......Page 387
W......Page 388
