Optimal control of the risk process in a regime-switching environment

This paper is concerned with cost optimization of an insurance company. The surplus of the insurance company is modeled by a controlled regime-switching diffusion, in which the regime-switching mechanism provides the fluctuations of the random environment. The goal is to find an optimal control that minimizes the total cost up to a stochastic exit time. A weaker sufficient condition than that of Fleming and Soner (2006, Section V.2) for the continuity of the value function is obtained. Further, the value function is shown to be a viscosity solution of a Hamilton–Jacobi–Bellman equation.