Controlled Markov Processes and Viscosity Solutions

by Wendell H. Fleming, Halil Mete Soner

Overview

This book is intended as an introduction to optimal shastic control for continuous time Markov processes and to the theory of viscosity solutions. Shastic control problems are treated using the dynamic programming approach. The authors approach shastic control problems by the method of dynamic programming. The fundamental equation of dynamic programming is a nonlinear evolution equation for the value function. For controlled Markov diffusion processes, this becomes a nonlinear partial differential equation of second order, called a Hamilton-Jacobi-Bellman (HJB) equation. Typically, the value function is not smooth enough to satisfy the HJB equation in a classical sense. Viscosity solutions provide framework in which to study HJB equations, and to prove continuous dependence of solutions on problem data. The theory is illustrated by applications from engineering, management science, and financial economics.

In this second edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.

Review of the earlier edition:

"This book is highly recommended to anyone who wishes to learn the dinamic principle applied to optimal shastic control for diffusion processes. Without any doubt, this is a fine book and most likely it is going to become a classic on the area... ."

SIAM Review, 1994

Product Details

ISBN-13:	9780387260457
Publisher:	Springer New York
Publication date:	11/17/2005
Series:	Stochastic Modelling and Applied Probability , #25
Edition description:	Second Edition 2006
Pages:	429
Product dimensions:	6.10(w) x 9.25(h) x (d)

Table of Contents

Deterministic Optimal Control.- Viscosity Solutions.- Optimal Control of Markov Processes: Classical Solutions.- Controlled Markov Diffusions in—n.- Viscosity Solutions: Second-Order Case.- Logarithmic Transformations and Risk Sensitivity.- Singular Perturbations.- Singular Shastic Control.- Finite Difference Numerical Approximations.- Applications to Finance.- Differential Games.