The stochastic shortest-path problem for Markov chains with infinite state space with applications to nearest-neighbor lattice chains

The aim of this paper is to solve the basic stochastic shortest-path problem (SSPP) for Markov chains (MCs) with countable state space and then apply the results to a class of nearest-neighbor MCs on the lattice state space \(\mathbb Z \times \mathbb Z \) whose only moves are one step up, down, to the right or to the left. The objective is to control the MC, by suppressing certain moves, so as to minimize the expected time to reach a certain given target state. We characterize the optimal policies for SSPPs for general MCs with countably infinite state space, the main tool being a verification theorem for the value function, and give an algorithmic construction. Then we apply the results to a large class of examples: nearest-neighbor MCs for which the state space \(\mathbb Z \times \mathbb Z \) is split by a vertical line into two regions inside which the transition probabilities are the same for every state. We give a necessary and sufficient condition for the so-called distance-diminishing policy to be optimal. For the general case in which this condition does not hold we develop an explicit finite construction of an optimal policy.