Stochastic asset allocation and reinsurance game under contagious claims

In this paper, we consider a stochastic asset allocation and reinsurance game between two insurance companies with contagious claims, where the insurance claim of one insurer can simultaneously affect the claim intensities of itself and its competitor. This clustering feature of claims is modelled by a mutual-excitation Hawkes process with exponential decays. Furthermore, we assume that the management of the insurance company wants to maximise the expected utility of the relative difference between its terminal surplus and that of its competitor at a fixed time point. The Nash equilibrium strategies have been constructed by solving the Hamilton–Jacobi–Bellman equations, where the explicit formulas of the optimal allocation policies have been derived to be independent of the claim intensities. We also introduce an iterative scheme based on the Feynman–Kac formula to compute the optimal proportional reinsurance policies numerically, where the existence and uniqueness of the solution to the fixed point equation and the convergence of the iterative numerical algorithm are proved rigorously. Finally, numerical examples are presented to show the effect of claim intensities on the optimal controls.