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Stochastic control with application in insurance. (English) Zbl 1134.91024
Frittelli, Marco (ed.) et al., Stochastic methods in finance. Lectures given at the C.I.M.E.–E.M.S. summer school, held in Bressanone/Brixen, Italy, July 6–12, 2003. Berlin: Springer (ISBN 3-540-22953-1/pbk). Lecture Notes in Mathematics 1856, 127-164 (2004).

Insurance means the transfer of risk to an insurance company. So these companies have to control risk as it arises in their business, but also investment risks. Thus concepts and techniques of stochastic optimal control should be useful in these areas of business. This survey article attempts to outline applications of this mathematical area to insurance. Conversely it sketches interesting applications for mathematicians working in stochastic control.

This article covers: 1. Possible control variables and stochastic control 2. Optimal investment for insurers 3. Optimal reinsurance and new business 4. Asymptotic behaviour for value functions and strategies 5. Control problems with constraints: dividends and ruin

These problems are discussed via continuous time simplified models of insurance and finance. In this set up the insurance risk is modelled by a Lundberg or Sparre-Andersen risk process. Thus risk is essentially understood as probability of ruin. In a model with investment these risk equations are coupled to one or several stochastic differential equations describing the assets in terms of Brownian motions. For reinsurance the risk equation is modified appropriately. The resulting Markov processes are best studied via their infinitesimal generator. As an alternative to the risk of ruin stochastic optimal control might be applied to such models. A problem here is a suitable objective function, though Hipp chooses utility U, a concept of doubtful utility in economics and decision theory. Optimal control leads to the Hamilton-Jacobi-Bellman equation for the value function. These techniques are briefly discussed in conncetion with some explicit models from the literature. Solutions of the HJB equation are notoriously hard to come by. For some models, however, it is possible to devise an iteration scheme which does not only yield an existence proof, but possibly also an algorithm for computing the solution.

In this review the author has covered a wide and difficult area of mathematics, and it might whet the appetite to delve deeper into it. On the other hand the lack of motivation and explanation will leave most readers alone. For the actuary the infinitesimal generator, with $L{\Psi }\left(s\right)=0,s\ge 0$ or the way to set up the HJB equation are certain stumbling blocks. For the mathematicians the insurcance concepts and the vagueness of the objective function leave a certain bewilderment.

##### MSC:
 91B30 Risk theory, insurance 93E20 Optimal stochastic control (systems)
##### Keywords:
risk; investment; reinsurance