Explicit solutions for an optimal stock selling problem under a Markov chain model

This paper is concerned with explicit solutions for a classical optimal stock selling problem. In contrast to almost all market models treated in the literature, the underlying market is solely determined by a two-state Markov chain. Such Markov chain model is strikingly simple and yet appears capable capturing various market movements ranging from close-to-Brownian motion to no-so-Brownian ones. The purpose of this paper is to study the optimal selling rule under such a model and develop a set of analysis techniques useful for related optimal stopping problems. In this paper, the goal of the problem under consideration is to find an optimal stopping time to sell the stock so as to maximize an expected return. Explicit solutions to the associated variational inequalities are obtained. These solutions are given in terms of a set of threshold levels. Verification theorems are provided to justify their optimality. Finally, numerical examples are provided to illustrate the results.