Existence and uniqueness of solutions to the Bellman equation in stochastic dynamic programming

Abstract

In this paper, we develop a framework to analyze stochastic dynamic optimization problems in discrete time. We obtain new results about the existence and uniqueness of solutions to the Bellman equation through a notion of Banach contractions that generalizes known results for Banach and local contractions. We apply the results obtained to an endogenous growth model and compare our approach with other well‐known methods, such as the weighted contraction method, countable local contractions, and the Q‐transform.