Robust optimal proportional reinsurance and investment strategy for an insurer and a reinsurer with delay and jumps

Abstract

In this paper, we consider a robust optimal proportional reinsurance and investment problem in a model with delay and jumps for an insurer and a reinsurer, who are concerned about model uncertainty in model parameters and aim to develop the robust optimal reinsurance and investment strategy. The surplus process of the insurer is described by a diffusion model (which is an approximation of the classical Crmér-Lundberg model) and the insurer can purchase proportional reinsurance. Assume that the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset, respectively. Moreover, the insurer and the reinsurer can invest in different risky assets whose price processes are governed by the jump-diffusion models. Specially, the insurer's and the reinsurer's wealth processes are modeled by stochastic differential delay equations by introducing the performance-related capital inflow (or outflow). Taking both the insurer and the reinsurer into account, this paper's target is to maximize the minimal expected product of the insurer's and the reinsurer's exponential utilities of the combination of terminal wealth and average performance wealth. By applying the techniques of stochastic optimal control theory, we build the robust Hamilton-Jacobi-Bellman equation. Furthermore, we obtain the robust optimal strategy and the corresponding value function, which extends those for the existing models to slightly more general versions. Finally, we present some numerical examples to demonstrate the impacts of some model parameters on the optimal strategy.