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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 $T\left(t\right)$, $t\ge 0,$ be the surplus process. The optimal invested amount ${A}_{t}$, $t\ge 0,$ at time $t$ has the following properties: the amount of money ${A}_{t}=A\left(T\left(t\right)\right)$; $A\left(0\right)=0$; the derivative ${A}^{\text{'}}$ 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.

##### MSC:
 91B30 Risk theory, insurance 93E20 Optimal stochastic control (systems) 60G40 Stopping times; optimal stopping problems; gambling theory