Stochastic Calculus of Variations: For Jump Processes

Description

This monograph is a concise introduction to the stochastic calculus of variations (also known as Malliavin calculus) for processes with jumps. It is written for researchers and graduate students who are interested in Malliavin calculus for jump processes. In this book "processes with jumps" includes both pure jump processes and jump-diffusions. The author provides many results on this topic in a self-contained way; this also applies to stochastic differential equations (SDEs) "with jumps". The book also contains some applications of the stochastic calculus for processes with jumps to the control theory and mathematical finance. Namely, asymptotic expansions functionals related with financial assets of jump-diffusion are provided based on the theory of asymptotic expansion on the Wiener–Poisson space. Solving the Hamilton–Jacobi–Bellman (HJB) equation of integro-differential type is related with solving the classical Merton problem and the Ramsey theory. The field of jump processes is nowadays quite wide-ranging, from the Lévy processes to SDEs with jumps. Recent developments in stochastic analysis have enabled us to express various results in a compact form. Up to now, these topics were rarely discussed in a monograph. Contents: Preface Preface to the second edition Introduction Lévy processes and Itô calculus Perturbations and properties of the probability law Analysis of Wiener–Poisson functionals Applications Appendix Bibliography List of symbols Index Preface Preface to the second edition Contents 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 for 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 previousmethods 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 d = 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 3.6.5 More on the density 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