Stochastic Calculus of Variations

For Jump Processes

Yasushi Ishikawa

• Publisher: De Gruyter
• Year: 2016
• eBook ISBN: 9783110378078
• Edition: 2
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

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

Table of contents

• Preface
• Preface to the second edition
• Contents
• Introduction
• 1. Lévy processes and Itô calculus
• 2. Perturbations and properties of the probability law
• 3. Analysis of Wiener–Poisson functionals
• 4. Applications
• Appendix
• Bibliography
• List of symbols
• Index