PROBABILITY AND ITS APPLICATIONS: DIFFUSIONS AND ELLIPTIC OPERATORS
Chapter 1 Integration by Parts and Absolute Continuity of Probability Laws 1 Chapter 2 Finite Dimensional Malliavin Calculus 7 21 The Ornstein-Uhlenbeck operator 7 22 The adjoint of the differential 12 23 An integration by parts formula: Existence of a density 13 24 Exercises 16 Chapter 3 The Basic Operators of Malliavin Calculus 17 31 The Ornstein-Uhlenbeck operator 18 32 The derivative operator 22 33 The integral or divergence operator 26 34 Differential calculus 27 35 Calculus with multiple Wiener integrals 33 36 Local property of the operators 39 37 Exercises 41 Chapter 4 Representation of Wiener Functionals 45 41 The Itˆo integral and the divergence operator 46 42 The Clark-Ocone formula 48 43 Generalized Clark-Ocone formula 49 44 Application to option pricing 54 45 Exercises 59 Chapter 5 Criteria for Absolute Continuity and Smoothness of Probability Laws 61 51 Existence of a density 61 52 Smoothness of the density 66 Chapter 6 Stochastic Partial Differential Equations Driven by Spatially Homogeneous Gaussian Noise 69 61 Stochastic integration with respect to coloured noise 69 62 Stochastic partial differential equations driven driven by a coloured noise 79 63 Exercises 90 Chapter 7 Malliavin Regularity of Solutions of SPDE’s 93 71 Exercises 120 Chapter 8 Analysis of the Malliavin Matrix of Solutions of SPDE’s 121 81 One dimensional case 121 82 Examples 135 83 Multidimensional case 146 Definitions of spaces 153