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**Controlled Markov processes and viscosity solutions.
2nd ed.**
*(English)*
Zbl 1105.60005

Stochastic Modelling and Applied Probability 25. New York, NY: Springer (ISBN 0-387-26045-5/hbk; 978-1-4419-2078-2/pbk; 0-387-31071-1/ebook). xvii, 428 p. (2006).

This classical book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions of Hamilton-Jacobi-Bellman partial differential equations. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions.

A new Chapter 10 gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter 6 of the first edition (1993; Zbl 0773.60070) has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter 11 gives a concise introduction to two-controller zero-sum differential games. Controlled Markov diffusion processes and viscosity solutions of Hamilton-Jacobi-Bellman partial differential equations are also covered. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g., large deviations theory) and with applications to engineering, physics, management, and finance. 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.

A new Chapter 10 gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter 6 of the first edition (1993; Zbl 0773.60070) has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter 11 gives a concise introduction to two-controller zero-sum differential games. Controlled Markov diffusion processes and viscosity solutions of Hamilton-Jacobi-Bellman partial differential equations are also covered. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g., large deviations theory) and with applications to engineering, physics, management, and finance. 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.

Reviewer: Pavel Gapeev (Berlin)

### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60J25 | Continuous-time Markov processes on general state spaces |

93E20 | Optimal stochastic control |

91A05 | 2-person games |

91A23 | Differential games (aspects of game theory) |

91G80 | Financial applications of other theories |