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**Stochastic optimization in insurance. A dynamic programming approach.**
*(English)*
Zbl 1308.91004

SpringerBriefs in Quantitative Finance. New York, NY: Springer (ISBN 978-1-4939-0994-0/pbk; 978-1-4939-0995-7/ebook). x, 146 p. (2014).

This book mainly contains work done by the authors during the last few years in the area of optimal control of insurance surpluses. The main problems are the minimisation of ruin probabilities by reinsurance and the maximisation of discounted dividend payments. Thus, the content of the book is similar to [the reviewer, Stochastic control in insurance. London: Springer (2008; Zbl 1133.93002)]. The methods, however, are different. The authors use the viscosity approach, which circumvents the problem that the value function (in the case of investment in a Black-Scholes market) may not possess second derivatives.

The book starts with an introduction to ruin theory and to dividends. In the second chapter, the optimisation problems are introduced. Basic properties and the Hamilton-Jacobi-Bellman (HJB) equation are found. Viscosity solutions are introduced in Chapter 3, and it is verified that the value functions are viscosity solutions to the HJB equation. In the next chapter, the value function is characterised as the minimal solution to the HJB equation. The optimal strategies are characterised in Chapter 5. In the last chapter, the problem of the numerical calculation of the value functions is investigated. In particular, it is shown that for Gamma distributed claim sizes, the value function of the optimal dividend problem not always is differentiable.

The book is very nicely written and gives an excellent overview of the topic. It is an ideal textbook for all researchers in insurance, in particular for those interested in optimisation problems.

The book starts with an introduction to ruin theory and to dividends. In the second chapter, the optimisation problems are introduced. Basic properties and the Hamilton-Jacobi-Bellman (HJB) equation are found. Viscosity solutions are introduced in Chapter 3, and it is verified that the value functions are viscosity solutions to the HJB equation. In the next chapter, the value function is characterised as the minimal solution to the HJB equation. The optimal strategies are characterised in Chapter 5. In the last chapter, the problem of the numerical calculation of the value functions is investigated. In particular, it is shown that for Gamma distributed claim sizes, the value function of the optimal dividend problem not always is differentiable.

The book is very nicely written and gives an excellent overview of the topic. It is an ideal textbook for all researchers in insurance, in particular for those interested in optimisation problems.

Reviewer: Hanspeter Schmidli (Köln)

### MSC:

91-02 | Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance |

91B30 | Risk theory, insurance (MSC2010) |

93E03 | Stochastic systems in control theory (general) |

93E20 | Optimal stochastic control |

49L20 | Dynamic programming in optimal control and differential games |

49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |

49N90 | Applications of optimal control and differential games |

91G80 | Financial applications of other theories |