Malliavi随机分析和相关论题

Introduction

1 Analysis on the Wiener space

1.1 Wiener chaos and stochastic integrals

1.1.1 The Wiener chaos decomposition

1.1.2 The white noise case: Multiple Wiener-Ito integrals

1.1.3 I to stochastic calculus

1.2 The derivative operator

1.2.1 The derivative operator in the white noise case

1.3 The divergence operator

1.3.1 Properties of the divergence operator

1.3.2 The Skorohod integral

1.3.3 The Ito stochastic integral as a particular case of the Skorohod integral

1.3.4 Stochastic integral representation of Wiener functionals

1.3.5 Local properties

1.4 The Ornstein-Uhlenbeck semigroup

1.4.1 The semigroup of Ornstein-Uhlenbeck

1.4.2 The generator of the Ornstein-Uhlenbeck semigroup

1.4.3 Hypercontractivity property and the multiplier theorem

1.5 Sobolev spaces and the equivalence of norms

2 Regularity of probability laws

2.1 Regularity of densities and related topics

2.1.1 Computation and estimation of probability densities

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

2.1.3 Absolute continuity using Bouleau and Hirsch's ap proach

2.1.4 Smoothness of densities

2.1.5 Composition of tempered distributions with nonde generate random vectors

2.1.6 Properties of the support of the law

2.1.7 Regularity of the law of the maximum of continuous processes

2.2 Stochastic differential equations

2.2.1 Existence and uniqueness of solutions

2.2.2 Weak differentiability of the solution

2.3 Hypoellipticity and Hormander's theorem

2.3.1 Absolute continuity in the case of Lipschitz coefficients

2.3.2 Absolute continuity under Hormander's conditions

2.3.3 Smoothness of the density under Hormander's condition

2.4 Stochastic partial differential equations

2.4.1 Stochastic integral equations on the plane

2.4.2 Absolute continuity for solutions to the stochastic heat equation

3 Anticipating stochastic calculus

3.1 Approximation of stochastic integrals

3.1.1 Stochastic integrals defined by Riemanns

3.1.2 The approach based on the L2 developme of the process

3.2 Stochastic calculus for anticipating integrals

3.2.1 Skorohod integral processes

3.2.2 Continuity and quadratic variation of the Skorohod integral

3.2.3 I to's formula for the Skorohod and Stratonovich integrals

3.2.4 Substitution formulas

3.3 Anticipating stochastic differential equations

3.3.1 Stochastic differential equations in the Sratonovich sense

3.3.2 Stochastic differential equations with boundary con ditions

……

4 Transformations of the Wiener measure

5 Fractional Brownian motion

6 Malliavin Calculus in finance

A Appendix

References

Index

There have been ten years since the publication of the first edition of this book. Since then， new applications and developments of the Malliavin cal- culus have appeared. In preparing this second edition we have taken into account some of these new applications， and in this spirit， the book has two additional chapters that deal with the following two topics: FYactional Brownian motion and Mathematical Finance.