Applied stochastic control of jump diffusions. 2nd ed. (English) Zbl 1116.93004

Øksendal, Bernt; Sulem, Agnès

Universitext. Berlin: Springer (ISBN 978-3-540-69825-8/pbk). xiii, 257 p. (2007).

The main purpose of this excellent monograph is to give a rigorous non-technical introduction to the most important and useful solution methods of various types of optimal stochastic control problems for jump diffusions and their applications. The covered types of control problems include classical stochastic control, optimal stopping, impulse control, and singular control. Both the dynamic programming method and the maximum principle method are discussed and relations between them are studied. Corresponding verification theorems involving the Hamilton-Jacobi-Bellman equation and/or (quasi-)variational inequalities are formulated. Viscosity solution formulation and numerical methods are also discussed. The text emphasises the applied aspect of the theory, mostly the applications to finance are discussed.
All the main results are illustrated by examples, and exercises appearing at the end of each chapter are accompanied with complete detailed solutions. This really helps the reader to understand the theory and to see how it can be applied. The book assumes some basic knowledge of stochastic analysis, measure theory, and partial differential equations. In this second edition of the book (see Zbl 1074.93009 for the first edition) a new chapter (Chapter 10) on optimal control of stochastic partial differential equations driven by Lévy processes is added. There is also a new section (Section 2.3) on optimal stopping with delayed information. Moreover, corrections and other improvements have beed made.

Reviewer: Pavel Gapeev (Berlin)

MSC:

93-02	Research exposition (monographs, survey articles) pertaining to systems and control theory
93E20	Optimal stochastic control
60-02	Research exposition (monographs, survey articles) pertaining to probability theory
60G40	Stopping times; optimal stopping problems; gambling theory
60J25	Continuous-time Markov processes on general state spaces
60J60	Diffusion processes
60J75	Jump processes (MSC2010)
60G51	Processes with independent increments; Lévy processes
49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
47J20	Variational and other types of inequalities involving nonlinear operators (general)
49J40	Variational inequalities
91G80	Financial applications of other theories

Keywords:

optimal stochastic control; optimal stopping; impulse control; singular control; jump-diffusion processes; stochastic (partial) differential equations; dynamic programming; maximum principle; Hamilton-Jacobi-Bellman equation; Hamilton-Jacobi-Bellman inequality; quasi-variational inequality; viscosity solutions; verification theorems; delayed information; applications to finance