Optimal Investment and Reinsurance Under the Gamma Process

In this paper, the insurance company invests its wealth in a capital market composed of a riskless asset and a risky asset. The aggregate claim process of the insurance company is modeled by the Gamma process so as to make it closer to the reality. In practice, the insurance company provides not only those policies with large lose coverings but also policies with small ones. The Gamma process can describe this characteristic better than the compound Poisson process. It is assumed that the insurance company can purchase proportional reinsurance or excess-of-loss reinsurance. Two kinds of optimization problems are considered: maximizing the expected utility of the terminal wealth and minimizing the probability of ruin. For the problem of maximizing the expected utility of the terminal wealth, the explicit optimal value functions and optimal strategies are obtained. For the problem of minimizing the ruin probability, a sufficient condition for the optimal value function and the optimal strategy is obtained.