Optimal Reinsurance and Investment Strategy for an Insurer in a Model with Delay and Jumps

This paper studies an optimal excess-of-loss reinsurance and investment problem in a model with delay and jumps for an insurer, who can purchase excess-of-loss reinsurance and invest his surplus in a risk-free asset and a risky asset whose price is governed by a jump-diffusion model. The insurer's surplus is described by a diffusion model, which is an approximation of the classical compound Poisson risk model. In particular, the wealth process of the insurer is modeled by a stochastic differential delay equation via introducing the performance-related capital inflow or outflow. Under the criterion for maximizing the expected exponential utility of the combination of terminal wealth and average performance wealth, optimal strategy and the corresponding value function are obtained by using the dynamic programming approach. Finally, numerical examples are provided to show the effects of model parameters on the optimal strategies and illustrate the economic meaning.