Optimal time-consistent reinsurance and investment strategies for a jump–diffusion financial market without cash

This paper considers an optimal reinsurance and investment strategies for an insurer under mean–variance criterion within a game theoretic framework. Specially, it is assumed that the surplus process is governed by a Cramér–Lundberg model, and apart from purchasing reinsurance, the insurer is allowed to invest in a financial market with multiple assets that all can be risky, whose price processes are modeled by the jump–diffusion process. Due to the market without cash, the method of separating the variables is not viable any more. We turn to an alternative approach to solve the extended Hamilton–Jacobi–Bellman equation, and closed-form expressions of the optimal strategies and value function are not only derived but also proved to be uniqueness. Moreover, some special cases of our model are provided and several numerical analyses for our results are presented as well. Under this criterion, different from existing literature, we find that (i) the value function is not linear but quadratic with respect to the current wealth; (ii) the optimal reinsurance and investment strategies depend on the wealth process; (iii) the parameters of risky assets(insurance market) have impacts on the optimal reinsurance(investment) policy; (iv) the safety loading of the insurer affects the optimal strategies.