## 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 $$T(t)$$, $$t \geq 0,$$ be the surplus process. The optimal invested amount $$A_t$$, $$t \geq 0,$$ 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.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 93E20 Optimal stochastic control 60G40 Stopping times; optimal stopping problems; gambling theory

Zbl 0846.90012
Full Text:

### References:

 [1] Browne, S., Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin, Mathematics of operations research, 20, 937-958, (1995) · Zbl 0846.90012 [2] Fleming, W.H., Soner, M., 1993. Controlled Markov Processes and Viscosity Solutions. Springer, New York. · Zbl 0773.60070 [3] Hipp, C., Taksar, M., 2000. Stochastic control for optimal new business. Insurance: Mathematics and Economics 26, 185-192. · Zbl 1103.91366 [4] Hoejgaard, B., Taksar, M., 1998. Optimal proportional reinsurance policies for diffusion models. Scandinavian Actuarial Journal, 166-180. · Zbl 1075.91559 [5] Schmidli, H., 1999. Optimal proportional reinsurance policies in a dynamic setting. Research Report 403. Department of Theoretical Statistics, Aarhus University, Denmark. · Zbl 0971.91039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.