Optimization Problems of Excess-of-Loss Reinsurance and Investment under the CEV Model

We consider that the insurer purchases excess-of-loss reinsurance and invests its wealth in the constant elasticity of variance (CEV) stock market. We model risk process by Brownian motion with drift and study the optimization problem of maximizing the exponential utility of terminal wealth under the controls of excess-of-loss reinsurance and investment. Using stochastic control theory and power transformation technique, we obtain explicit expressions for the optimal polices and value function. We also show that the optimal excess-of-loss reinsurance is always better than optimal proportional reinsurance. Some numerical examples are given. 1. Introduction Many papers deal with optimal reinsurance or optimal investment issues for diffusion approximation risk models in the past ten years. In these papers, the insurer is allowed to take reinsurance and/or invest its capital in the Black-Scholes market. Some of the problems have been dealt with through stochastic control theory and related methodologies for finding the minimum probability of ruin or the maximum expected utility of terminal wealth. Browne [1] used a Brownian motion with a drift to describe the surplus of the insurer and found the optimal investment policy to maximize the expected exponential utility of terminal wealth. Later, Schmidli [2], Taksar and Markussen [3] considered the optimal reinsurance policy which minimizes the ruin probability of the cedent. Recently, much research on insurance optimization in the presence of both proportional reinsurance and investment has been done. Luo et al. [4] studied optimal proportional reinsurance and investment policy which minimizes the probability of ruin. Bai and Guo [5] investigated the problem of maximizing the expected exponential utility of terminal wealth with multiple risky assets and proportional reinsurance. For related works, see, for example, Promislow and Young [6], Liang and Guo [7] and references therein. The excess-of-loss reinsurance has also attracted interest among academia and practitioners. Asmussen et al. [8] studied a dynamic choice of excess-of-loss reinsurance retention level and the dividend distribution policy which maximizes the expected present value of the dividends in a diffusion model. Zhang et al. [9] considered the problem of minimizing the probability of ruin by controlling the combinational quota-share and excess-of-loss reinsurance strategy. Meng and Zhang [10] investigated optimal risk control for the excess-of-loss reinsurance policy which minimizes the probability of ruin. Bai and Guo [11] explored optimal