Malliavin Calculus With Applications to Stochastic Partial Differential Equations

by Sanz-Sole; Marta

• ISBN13: 9780849340307
• ISBN10: 0849340306
• Format: Hardcover
• Copyright: 2005-08-17
• Publisher: EFPL Press

Summary

Developed in the 1970s to study the existence and smoothness of density for the probability laws of random vectors, Malliavin calculus--a stochastic calculus of variation on the Wiener space--has proven fruitful in many problems in probability theory, particularly in probabilistic numerical methods in financial mathematics.This book presents applications of Malliavin calculus to the analysis of probability laws of solutions to stochastic partial differential equations driven by Gaussian noises that are white in time and coloured in space. The first five chapters introduce the calculus itself based on a general Gaussian space, going from the simple, finite-dimensional setting to the infinite-dimensional one. The final three chapters discuss recent research on regularity of the solution of stochastic partial differential equations and the existence and smoothness of their probability laws.About the author:Marta Sanz-Solé is Professor at the Faculty of Mathematics, University of Barcelona. She is a leading member of the research group on stochastic analysis at Barcelona, and in 1998 she received the Narcis Monturiol Award of Scientific and Technological Excellence from the autonomous government of Catalonia.

Table of Contents

Introduction	v
CHAPTER 1 Integration by Parts and Absolute Continuity of Probability Laws	1	(6)
CHAPTER 2 Finite Dimensional Malliavin Calculus	7	(10)
2.1 The Ornstein-Uhlenbeck operator	7	(5)
2.2 The adjoint of the differential	12	(1)
2.3 An integration by parts formula: Existence of a density	13	(3)
2.4 Exercises	16	(1)
CHAPTER 3 The Basic Operators of Malliavin Calculus	17	(28)
3.1 The Ornstein-Uhlenbeck operator	18	(4)
3.2 The derivative operator	22	(4)
3.3 The integral or divergence operator	26	(1)
3.4 Differential calculus	27	(6)
3.5 Calculus with multiple Wiener integrals	33	(6)
3.6 Local property of the operators	39	(2)
3.7 Exercises	41	(4)
CHAPTER 4 Representation of Wiener Functionals	45	(16)
4.1 The Itô integral and the divergence operator	46	(2)
4.2 The Clark-Ocone formula	48	(1)
4.3 Generalized Clark-Ocone formula	49	(5)
4.4 Application to option pricing	54	(5)
4.5 Exercises	59	(2)
CHAPTER 5 Criteria for Absolute Continuity and Smoothness of Probability Laws	61	(8)
5.1 Existence of a density	61	(5)
5.2 Smoothness of the density	66	(3)
CHAPTER 6 Stochastic Partial Differential Equations Driven by Spatially Homogeneous Gaussian Noise	69	(24)
6.1 Stochastic integration with respect to coloured noise	69	(10)
6.2 Stochastic partial differential equations driven driven by a coloured noise	79	(11)
6.3 Exercises	90	(3)
CHAPTER 7 Malliavin Regularity of Solutions of SPDE's	93	(28)
7.1 Exercises	120	(1)
CHAPTER 8 Analysis of the Malliavin Matrix of Solutions of SPDE's	121	(32)
8.1 One dimensional case	121	(14)
8.2 Examples	135	(11)
8.3 Multidimensional case	146	(7)
Definitions of spaces	153	(2)
Bibliography	155