Optimal proportional reinsurance policies for stochastic models

Abstract

In order to minimize the insurance risk, the insurer can transfer a part of the whole risk to a second insurer, called reinsurer, using the mechanism of reinsurance. The latest must be chosen carefully because ceding huge risk implies ceding huge amount of premiums and the inverse case may affect the financial situation of the first insurer and can cause bankruptcy. The question is how to find an optimal choice of reinsurance? This task may be simple when we deal with a static strategy and this problem was treated carefully in the actuarial literature. But, when we’d like to optimize reinsurance strategy at any time $t\ge 0,$ this task becomes more complicated. We assume a diffusion model for the insurer’s surplus. Stochastic control theory gives a variety of solutions to the last problem when adopting ruin’s probability minimizing criterion. In the present paper, we treat the case of proportional reinsurance that we optimize subject to maximizing the cedent embedded value. To achieve this goal, we consider a problem of optimal reinsurance and investment for an insurance company whose surplus is governed by a linear diffusion model.

Keywords:

• Proportional reinsurance
• investment
• time of ruin
• stochastic control
• embedded value
• Hamilton–Jacobi–Bellman equation

Additional information

Funding

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program [grant number R.G.P.1/129/40].