Optimal excess-of-loss reinsurance and investment problem with delay and jump-diffusion risk process under the CEV model

In this paper, we investigate an optimal investment and excess-of-loss reinsurance problem with delay and jump-diffusion risk process for an insurer. Specifically, the insurer is allowed to purchase excess-of-loss reinsurance and invest in a financial market, where the surplus of insurer is represented by a jump-diffusion model and the financial market consists of one risk-free asset and one risky asset whose price process is governed by a constant elasticity of variance model. In addition, the performance-related capital inflow/outflow is introduced, the wealth process of insurer is modeled by a stochastic differential delay equation. The insurer aims to seek the optimal excess-of-loss reinsurance and investment strategy to maximize the expected exponential utility of the combination of terminal wealth and average performance wealth. By solving a Hamilton-Jacobi-Bellman equation, the closed-form expressions for the optimal strategy and the optimal value function are derived. Finally, some special cases of our model and results are presented, and some numerical examples for our results are provided.