Malliavin Calculus with Applications to SPDE

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Marta Sanz-Solé,2005, CRC Press.

Contents

Introduction .......................................................v
Chapter 1 Integration by Parts and Absolute Continuity
of Probability Laws 1
Chapter 2 Finite Dimensional Malliavin Calculus 7
2.1 The Ornstein-Uhlenbeck operator . . . . . . . . . . . . . . . . . . 7
2.2 The adjoint of the differential . . . . . . . . . . . . . . . . . . . . .12
2.3 An integration by parts formula:
Existence of a density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Chapter 3 The Basic Operators of Malliavin Calculus 17
3.1 The Ornstein-Uhlenbeck operator. . . . . . . . . . . . . . . . .18
3.2 The derivative operator. . . . . . . . . . . . . . . . . . . . . . . . . . .22
3.3 The integral or divergence operator . . . . . . . . . . . . . . . 26
3.4 Differential calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
3.5 Calculus with multiple Wiener integrals. . . . . . . . . . .33
3.6 Local property of the operators . . . . . . . . . . . . . . . . . . . 39
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Chapter 4 Representation of Wiener Functionals 45
4.1 The Itˆo integral and the divergence operator . . . . . .46
4.2 The Clark-Ocone formula. . . . . . . . . . . . . . . . . . . . . . . . .48
4.3 Generalized Clark-Ocone formula . . . . . . . . . . . . . . . . . 49
4.4 Application to option pricing . . . . . . . . . . . . . . . . . . . . . 54
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Chapter 5 Criteria for Absolute Continuity
and Smoothness of Probability Laws 61
5.1 Existence of a density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Smoothness of the density . . . . . . . . . . . . . . . . . . . . . . . . 66
© 2005, First edition, EPFL Press
viii Contents
Chapter 6 Stochastic Partial Differential Equations
Driven by Spatially Homogeneous Gaussian Noise 69
6.1 Stochastic integration
with respect to coloured noise . . . . . . . . . . . . . . . . . . . . 69
6.2 Stochastic partial differential equations
driven driven by a coloured noise . . . . . . . . . . . . . . . . . 79
6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Chapter 7 Malliavin Regularity of Solutions of SPDE’s 93
7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Chapter 8 Analysis of the Malliavin Matrix of Solutions
of SPDE’s 121
8.1 One dimensional case. . . . . . . . . . . . . . . . . . . . . . . . . . . .121
8.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.3 Multidimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Definitions of spaces 153
Bibliography 155
© 2005,

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