×

Applied stochastic control of jump diffusions. (English) Zbl 1074.93009

Universitext. Berlin: Springer (ISBN 3-540-14023-9/pbk). x, 208 p. (2005).
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.
The book contains ten chapters, and the last one is devoted to the solutions of exercises. Chapter 1 presents a brief review of the basic concepts and results needed for the applied calculus of jump-diffusion processes (or jump diffusions) solving stochastic differential equations driven by Lévy processes, and their applications to models in finance. In Chapter 2, a general formulation of the optimal stopping problem for jump diffusions is studied. The corresponding integro-variational inequalities are presented, and the verification theorem is proved. The results are illustrated with the problem of finding an optimal stopping time when one should sell a financial asset modelled by a geometric Lévy process.
In Chapter 3, optimal stochastic control problems for jump diffusions are studied. The corresponding dynamic programming equation for the jump-diffusion case being an analogue of the Hamilton-Jacobi-Bellman equation for the continuous case is presented, and the verification theorem is proved. As an illustrating example, the problem of finding optimal consumption and portfolio (solved by Merton in the continuous case) in a Lévy-type Black-Scholes market is considered. A sufficient maximum principle is also presented, and its applications to finance are discussed. In Chapter 4, combined optimal stopping and stochastic control problems and, associated with them, Hamilton-Jacobi-Bellman variational inequalities are discussed in a jump-diffusion setting. This chapter may also serve as a brief review of the theory of optimal stopping and its variational inequalities on one hand, and the theory of stochastic control and its Hamilton-Jacobi-Bellman equations on the other. As an illustration of how combined optimal stopping and stochastic control may appear in economics, an optimal resource extraction control and stopping problem is considered.
In Chapter 5, a general theory of singular control of jump diffusions is given. The corresponding integro-variational inequalities are presented, and the related verification theorem is proved. In order to illustrate singular control problems, the problem of optimal consumption rate under proportional transaction costs is considered. In Chapter 6, optimal impulse control problems for jump diffusions are studied. The corresponding quasi-integrovariational inequalities are presented, and the related verification theorem is proved. The results are illustrated with the problem of optimal stream of dividends under transaction costs in a jump-diffusion market model.
Chapter 7 studies the problem of approximating optimal impulse control of jump diffusions by sequences of iterated optimal stopping problems. The corresponding iterative scheme, which makes it possible to reduce a given impulse control problem to a sequence of iterated optimal stopping problems, is given. This schema turns out to be a useful tool both for theoretical purposes and numerical applications. An illustrating example is also given. In Chapter 8, a combined stochastic control and impulse control problem is studied in a jump-diffusion framework. In addition to the general situation considered in Chapter 6, here one is free to choose a Markovian control at any state. The corresponding Hamilton-Jacobi-Bellman quasi-integro-variational inequalities are given and the verification theorem is proved. As examples, the problems of optimal combined control of the exchange rate and of optimal consumption and portfolio with both fixed and proportional transaction costs are considered. As an approximation, an iterated procedure for a combined stochastic control and impulse control by iterated combined stochastic control and optimal stopping is presented. The presentation of this method is similar to the approach in Chapter 7.
In Chapter 9, the authors consider viscosity solutions to the (quasi-)variational inequalities appearing in the previous sections. The nice feature of viscosity solutions is that they also apply to the nonlinear equations appearing in control theory. At the end of the chapter, some insights to the corresponding numerical analysis of Hamilton-Jacobi-Bellman (quasi-)variational inequalities based on finite-difference approximations are given. In Chapter 10, solutions to selected exercises are presented.
This book is a very useful text for students, researchers, and practitioners working in stochastic analysis and its applications.

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
91B28 Finance etc. (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI