Optimal stopping problem for jump–diffusion processes with regime-switching

Abstract

This paper concerns the optimal stopping problem in an infinite horizon for jump–diffusion processes with regime-switching. It is found that the jumps of the studied process have an important impact on the existence of the optimal stopping times. In this work we provide a sufficient condition on the jumps of the process in terms of the gain function to ensure the existence of the optimal stopping times, which is shown to be quite sharp by an illustrative example. Additionally, an explicit representation of the ε-optimal stopping time is given. In order to characterize the associated value function, we show that it is a unique viscosity solution to a coupled system of Hamilton–Jacobi–Bellman equations. In the meanwhile, we unify two existing definitions of viscosity solutions for the Hamilton–Jacobi–Bellman equations associated with the regime-switching processes.