The Malliavin calculus (or stochastic calculus of variations) is an infinite-dimensional differential calculus on a Gaussian space. Originally, it was developed to provide a probabilistic proof to H?rmander's "sum of squares" theorem, but it has found a wide range of applications in stochastic analysis. This monograph presents the main features of the Malliavin calculus and discusses in detail its main applications. The author begins by developing the analysis on the Wiener space, and then uses this to establish the regularity of probability laws and to prove H?rmander's theorem. The regularity of the law of stochastic partial differential equations driven by a space-time white noise is also studied. The subsequent chapters develop the connection of the Malliavin with the anticipating stochastic calculus, studying anticipating stochastic differential equations and the Markov propertynbsp;of solutions to stochastic differential equations with boundary conditions.The second edition of this monograph includes recent applications of the Malliavin calculus in finance and a chapter devoted to the stochastic calculus with respect to the fractional Brownian motion.

The Malliavin Calculus and Related Topics
作者：David Nualart
Edition: 2, revised
由Springer出版, 2006
ISBN 038794432X, 9780387944326
382 页

*大小：426页，3.21M
*格式：PDF。

内容：

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?o integrals . 8
1.1.3 It?o 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?o 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 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 It?o’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

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

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