Malliavin Calculus With Applications to

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.

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

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Malliavin Calculus With Applications to Stochastic Partial Differential Equations

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