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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 $T(t)$, $t \geq 0,$ 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, {\it 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
Full Text: DOI
[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