Stochastic dynamic programming under recursive Epstein-Zin preferences

TL;DR

This paper establishes the existence and uniqueness of solutions to the Bellman equation in stochastic dynamic programming with recursive Epstein-Zin preferences, using weighted norms and fixed point theorems, improving convergence bounds.

Contribution

It introduces a novel approach to solving Bellman equations with unbounded utilities, avoiding boundary conditions and enhancing convergence analysis.

Findings

Proves Bellman equation solutions via Banach fixed point theorem.

Provides better bounds for value iteration convergence.

Handles parameter ranges where Du's theorem fails.

Abstract

This paper investigates discrete-time Markov decision processes with recursive utilities (or payoffs) defined by the classic CES aggregator and the Kreps-Porteus certainty equivalent operator. According to the classification introduced by Marinacci and Montrucchio, some aggregators that we consider are Thompson and some of them are neither Thompson nor Blackwell. We focus on the existence and uniqueness of a solution to the Bellman equation. Since the per-period utilities can be unbounded, we work with the weighted supremum norm. Our paper shows three major points for such models. Firstly, we prove that the Bellman equation can be obtained by the Banach fixed point theorem for contraction mappings acting on a standard complete metric space. Secondly, we need not assume any boundary conditions, which are present when the Thompson metric or the Du's theorem are used. Thirdly, our results give better bounds for the geometric convergence of the value iteration algorithm than those obtained by Du's fixed point theorem. Moreover, our techniques allow to derive the Bellman equation for some values of parameters in the CES aggregator and the Kreps-Porteus certainty equivalent that cannot be solved by Du's theorem for increasing and convex or concave operators acting on an ordered Banach space.