Constrained optimal stopping under a regime-switching model

TL;DR

This paper studies an optimal stopping problem for a regime-switching geometric Brownian motion with constraints on stopping times, deriving the value function, optimal thresholds, and analyzing asymptotic behaviors.

Contribution

It introduces a novel framework for constrained optimal stopping under regime-switching, providing explicit solutions and numerical analysis.

Findings

Optimal stopping time exists as a threshold under certain conditions.

Derived explicit expressions for value functions and thresholds.

Numerical results illustrate the theoretical findings.

Abstract

We investigate an optimal stopping problem for the expected value of a discounted payoff on a regime-switching geometric Brownian motion under two constraints on the possible stopping times: only at exogenous random times and only during a specific regime. The main objectives are to show that an optimal stopping time exists as a threshold type under some boundary conditions and to derive expressions of the value functions and the optimal threshold. To this end, we solve the corresponding variational inequality and show that its solution coincides with the value functions. Some numerical results are also introduced. Furthermore, we investigate some asymptotic behaviors.