Robust optimal stopping with regime switching

TL;DR

This paper develops a theoretical framework for robust optimal stopping problems under model uncertainty and regime switching, using dynamic programming, viscosity solutions, and singular perturbation methods, with applications to stock selling timing.

Contribution

It introduces a comprehensive approach combining viscosity solutions and singular perturbation techniques for robust optimal stopping with regime switching.

Findings

Characterized the value function as a viscosity solution to the HJB equation.

Proved the smooth-fit principle in the context of regime switching.

Provided an asymptotically optimal solution for large state space Markov chains.

Abstract

In this paper, we study an optimal stopping problem in the presence of model uncertainty and regime switching. The max-min formulation for robust control and the dynamic programming approach are adopted to establish a general theoretical framework for such kind of problem. First, based on the dynamic programming principle, the value function of the optimal stopping problem is characterized as the unique viscosity solution to the associated Hamilton-Jacobi-Bellman equation. Then, the so-called smooth-fit principle for optimal stopping problems is proved in the current context, and a verification theorem consisting of a set of sufficient conditions for robust optimality is established. Moreover, when the Markov chain has a large state space and exhibits a two-time-scale structure, a singular perturbation approach is utilized to reduce the complexity involved and obtain an asymptotically optimal solution. Finally, an example of choosing the best time to sell a stock is provided, in which numerical experiments are reported to illustrate the implications of model uncertainty and regime switching.