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 $\epsilon$-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.

本文涉及具有状态切换的跳跃扩散过程在无限范围内的最优停止问题。发现所研究过程的跳跃对最佳停止时间的存在具有重要影响。在这项工作中，就增益函数而言，我们为过程的跳跃提供了充分的条件，以确保存在最佳的停止时间，通过示例说明，该时间非常明显。此外，$\epsilon$-给出了最佳的停止时间。为了表征相关的值函数，我们表明它是汉密尔顿-雅各比-贝尔曼方程组耦合系统的唯一粘性解。同时，我们统一了与状态切换过程相关的Hamilton–Jacobi–Bellman方程的粘度解决方案的两个现有定义。