Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods

This work develops a stochastic differential game model between two insurance companies who adopt the optimal reinsurance strategies to reduce the risk. The surplus is modeled by a regime-switching jump diffusion process. A single payoff function is imposed, and one player devises an optimal strategy to maximize the expected payoff function, whereas the other player is trying to minimize the same quantity. Using dynamic programming principle, the upper and lower values of the game satisfy a coupled system of nonlinear integro-differential Hamilton-Jacobi-Isaacs (HJI) equations. Moreover, the existence of the saddle point for this game problem is verified. Because of the jumps and regime-switching, closed-form solutions are virtually impossible to obtain. Our effort is devoted to designing numerical methods. We use Markov chain approximation techniques to construct a discrete-time controlled Markov chain to approximate the value functions and optimal controls. Convergence of the approximation algorithms is proved. Examples are presented to illustrate the applicability of the numerical methods.