Optimal Switching Between Two Random Walks

Abstract

This paper is motivated by remarkable results of Mandelbaum, Shepp and Vanderbei concerning an optimal switching problem for two Brownian motions. In this paper, the discrete form of this problem, in which the Brownian motions are replaced by random walks, is studied and solved without any restriction on the boundary data. The method proposed here involves uncovering the structure of the solution using combinatorial and geometric arguments, and then providing a characterization for the two types of possible solutions, as well as explicit formulas for computing the solution. The extension of these methods and results to the continuous time problem will be considered in a subsequent paper.