Existence and Uniqueness of Solutions to the Stochastic Bellman Equation with Unbounded Shock

TL;DR

This paper establishes conditions for the existence and uniqueness of solutions to the stochastic Bellman equation with unbounded shocks, broadening the scope of dynamic economic models.

Contribution

It introduces a general fixed point theorem applicable to unbounded shocks, extending previous results for stochastic dynamic programming.

Findings

Proves existence and uniqueness of solutions under new conditions.

Applies results to endogenous growth and Lucas asset pricing models.

Expands applicability of stochastic dynamic models with unbounded shocks.

Abstract

In this paper we develop a general framework to analyze stochastic dynamic problems with unbounded utility functions and correlated and unbounded shocks. We obtain new results of the existence and uniqueness of solutions to the Bellman equation through a general fixed point theorem that generalizes known results for Banach contractions and local contractions. We study an endogenous growth model as well as the Lucas asset pricing model in an exchange economy, significantly expanding their range of applicability.