An algorithm based on an iterative optimal stopping method for Feller processes with applications to impulse control, perturbation, and possibly zero random discount problems

Abstract

In this paper we present an iterative optimal stopping method for general optimal stopping problems for Feller processes. We show using an approximating scheme that the value function of an optimal stopping problem for some general operator is the unique viscosity solution to an Hamilton–Jacobi–Bellman equation (see for example Theorems 2.3 and 2.4). We apply our results to study impulse control problems for Feller–Markov processes and derive explicit solutions in the case of one dimensional regular Feller diffusion. We also use our result to study optimal stopping problems for both regime switching and semi-Markov processes and characterize their value functions as the limit of iterative optimal stopping problems (see Corollary 4.2 and Proposition 4.3 ). Finally, we examine optimal stopping problems for random (possibly zero) discount.