Contents
1 Introduction 1
2 Integration by parts and absolute continuity of probability laws 3
3 Finite dimensional Malliavin calculus 7
3.1 The Ornstein-Uhlenbeck operator . . . . . . . . . . . . . . . . . . 7
3.2 The adjoint of the di®erential . . . . . . . . . . . . . . . . . . . . 11
3.3 An integration by parts formula: Existence of density . . . . . . 12
4 The basic operators of Malliavin Calculus 15
4.1 The Ornstein-Uhlenbeck operator . . . . . . . . . . . . . . . . . . 15
4.2 The derivative operator . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 The integral or divergence operator . . . . . . . . . . . . . . . . . 21
4.4 Di®erential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.5 Calculus with multiple Wiener integrals . . . . . . . . . . . . . . 26
4.6 Local property of the operators . . . . . . . . . . . . . . . . . . . 30
5 Representation of Wiener Functionals 34
5.1 The It^o integral and the divergence operator . . . . . . . . . . . 34
5.2 The Clark-Ocone formula . . . . . . . . . . . . . . . . . . . . . . 36
5.3 Generalized Clark-Ocone formula . . . . . . . . . . . . . . . . . . 37
5.4 Application to option pricing . . . . . . . . . . . . . . . . . . . . 40
6 Criteria for absolute continuity and smoothness of probability
laws 46
6.1 Existence of density . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.2 Smoothness of the density . . . . . . . . . . . . . . . . . . . . . . 49
7 Stochastic partial di®erential equations driven by a Gaussian
spatially homogeneous correlated noise 52
7.1 Stochastic integration with respect to coloured noise . . . . . . . 52
7.2 Stochastic Partial Di®erential Equations driven by a coloured noise 60
8 Malliavin regularity of solutions of SPDEs 72
9 Analysis of the Malliavin matrix of solutions of SPDEs 95
9.1 One dimensional case . . . . . . . . . . . . . . . . . . . . . . . . 95
9.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
9.3 Multidimensional case . . . . . . . . . . . . . . . . . . . . . . . . 115
10 De¯nition of spaces used along the course 120
References 121