Time-consistent reinsurance and investment strategy combining quota-share and excess of loss for mean-variance insurers with jump-diffusion price process

This article investigates an optimal time-consistent reinsurance and investment problem under the mean-variance criterion for an insurer whose surplus process is governed by a compound Poisson risk model. The insurer is allowed to purchase combining quota-share and excess of loss reinsurance for claims and to invest in financial market to increase his wealth. The financial market consists of a risk-free asset and a risky asset whose price process follows a jump-diffusion process. The insurer wants to find an optimal time-consistent reinsurance-investment strategy so as to maximize the expected terminal surplus while minimizing the variance of the terminal surplus. By using the stochastic control technique, we obtain the explicit solutions for the optimal reinsurance-investment strategy and the corresponding optimal value function. Finally, numerical experiments are carried out to illustrate the effects of model parameters on the optimal time-consistent reinsurance-investment strategy, and to compare the pure excess of loss reinsurance with the pure quota-share reinsurance through analyzing the optimal value function.