Differential Properties of Objective and Value Functions for Discounted Markov Decision Processes with Finite State and Action Sets

This paper studies differential properties of objective functions for discrete-time discounted Markov decision processes (MDPs) with finite state and action sets, when differentiation is considered with respect to the discount factor. Given a fixed discount factor for an MDP, we estimate the norms and seminorms of the objective functions, their derivatives, and the value function at a different discount factor. We then use these estimates to prove that objective functions, their derivatives, and value functions are Lipschitz continuous in discount factors. As a result, if an MDP is solved with a fixed discount factor, then we can estimate the value function at a different discount factor without solving it again.\\ %Following these estimations, we also reveal the relations between the seminorms of objective functions, their behavior when the discount factor approaches $1,$ and the discount factors at which the set of optimal policies changes.