Preface -- 0 Introduction -- 1 Lévy processes and Itô calculus -- 1.1 Poisson random measure and Lévy processes -- 1.1.1 Lévy processes -- 1.1.2 Examples of Lévy processes -- 1.1.3 Stochastic integral for a finite variation process -- 1.2 Basic materials to SDEs with jumps -- 1.2.1 Martingales and semimartingales -- 1.2.2 Stochastic integral with respect to semimartingales -- 1.2.3 Doléans' exponential and Girsanov transformation -- 1.3 Itô processes with jumps -- 2 Perturbations and properties of the probability law -- 2.1 Integration-by-parts on Poisson space -- 2.1.1 Bismut's method -- 2.1.2 Picard's method -- 2.1.3 Some previous methods -- 2.2 Methods of finding the asymptotic bounds (I) -- 2.2.1 Markov chain approximation -- 2.2.2 Proof of Theorem 2.3 -- 2.2.3 Proof of lemmas -- 2.3 Methods of finding the asymptotic bounds (II) -- 2.3.1 Polygonal geometry -- 2.3.2 Proof of Theorem 2.4 -- 2.3.3 Example of Theorem 2.4 - easy cases -- 2.4 Summary of short time asymptotic bounds -- 2.4.1 Case that µ(dz) is absolutely continuous with respect to the m-dimensional Lebesgue measure dz -- 2.4.2 Case that µ(dz) is singular with respect to dz -- 2.5 Auxiliary topics -- 2.5.1 Marcus'canonical processes -- 2.5.2 Absolute continuity of the infinitely divisible laws -- 2.5.3 Chain movement approximation -- 2.5.4 Support theorem for canonical processes -- 3 Analysis of Wiener-Poisson functionals -- 3.1 Calculus of functionals on the Wiener space -- 3.1.1 Definition of the Malliavin-Shigekawa derivative Dt -- 3.1.2 Adjoint operator δ = D* -- 3.2 Calculus of functionals on the Poisson space -- 3.2.1 One-dimensional case -- 3.2.2 Multidimensional case -- 3.2.3 Characterisation of the Poisson space -- 3.3 Sobolev space for functionals over the Wiener-Poisson space -- 3.3.1 The Wiener space -- 3.3.2 The Poisson Space.

3.3.3 The Wiener-Poisson space -- 3.4 Relation with the Malliavin operator -- 3.5 Composition on the Wiener-Poisson space (I) - general theory -- 3.5.1 Composition with an element in S' -- 3.5.2 Sufficient condition for the composition -- 3.6 Smoothness of the density for Itô processes -- 3.6.1 Preliminaries -- 3.6.2 Big perturbations -- 3.6.3 Concatenation (I) -- 3.6.4 Concatenation (II) - the case that (D) may fail -- 3.7 Composition on the Wiener-Poisson space (II) - Itô processes -- 4 Applications -- 4.1 Asymptotic expansion of the SDE -- 4.1.1 Analysis on the stochastic model -- 4.1.2 Asymptotic expansion of the density -- 4.1.3 Examples of asymptotic expansions -- 4.2 Optimal consumption problem -- 4.2.1 Setting of the optimal consumption -- 4.2.2 Viscosity solutions -- 4.2.3 Regularity of solutions -- 4.2.4 Optimal consumption -- 4.2.5 Historical sketch -- Appendix -- Bibliography -- List of symbols -- Index.