Jingxiao Zhang, Kai Cao, Kannan D.

【Abstract】

The optimal reinsurance and investment problem for insurance has attracted a lot of attention of researchers inthe field of stochastic control for a long time. Along this line we discuss this problem in the case of jump diffusionmarkets when neither short-selling nor borrowing is allowed. Here, we specifically assume that the risk process of the insurance company is a diffusion process. The insurance company can transfer its risk by reinsurance and also invest its surplus in the financial market, where we model the price of the risky asset by a geometric Levy process. To maximize the CARA (Constant Absolute Risk Aversion) utility of terminal wealth, the HJB equation with no short-selling constraint has been considered, and we obtain the closed form of the value function by a standard method. However, only a handful of people have discussed this problem under both constraints, (i.e. noshort-selling and no borrowing). This is because the problem is much more general in this context, and becomes so complex that analytical solution could hardly be obtained. Therefore, we provide, under the no short-sellingand no borrowing constraints, a numerical solution via Markov chain approximation, which proves to be effective and amenable.

【Keywords】

optimal reinsurance, jump-diffusion markets, HJB equation, Markov chain approximation, no short-selling and no borrowing, CARA utility function