Optimal mean-variance reinsurance with delay and multiple\\ classes of dependent risks

Abstract

In this paper, we consider an optimal proportional reinsurance problem for the compound Poisson risk model with delay and multiple dependent classes of insurance business. Under the criterion of maximizing the mean-variance utility of the terminal wealth, we formulate the time-inconsistent problem within a game theoretic framework and look for a subgame perfect Nash equilibrium strategy. Based on the technique of stochastic control theory and the corresponding extended Hamilton-Jacobi-Bellman equation, the explicit expressions of the optimal equilibrium strategy and the value function are derived for the case of insurance business $n=2$. Meanwhile, we derive the closed-form expressions of optimal solutions for the case of $n=3$ by the method of dimension reduction which can be used to get the optimal results for the case of $n>3$. Finally, some numerical examples are presented to show the impact of model parameters on the optimal strategies.