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Optimal investment for insurers. (English) Zbl 1007.91025

Insurance business considered is modelled by a compound Poisson process with the Black-Scholes type market index. The authors show that the ruin probability of this risk process is minimized by the choice of a suitable investment strategy for a capital market index. Let T(t), t0, be the surplus process. The optimal invested amount A t , t0, at time t has the following properties: the amount of money A t =A(T(t)); A(0)=0; the derivative A ' has a pole at 0; the function A remains bounded for exponential claim sizes, and it is unbounded for heavy-tailed claim size distributions. The result is obtained with the aid of the Bellman equation - a second order nonlinear integro-differential equation - which characterizes the value function and the optimal strategy. More explicit solutions are determined when the claim size distribution is exponential, in which case a numerical example is also provided. Another example refers to the case of Pareto claim size.

Using in the model a Brownian motion with drift in place of the compound Poisson process, S. Browne [Meth. Oper. Res. 20, 937-958 (1995; Zbl 0846.90012)] obtained the quite different result: the optimal strategy is the investment of a constant amount of money in the risky asset, irrespectively of the size of the surplus.

91B30Risk theory, insurance
93E20Optimal stochastic control (systems)
60G40Stopping times; optimal stopping problems; gambling theory