Stability of the Epstein-Zin problem

TL;DR

This paper analyzes how small changes in market dynamics affect the stability of the Epstein-Zin utility optimization problem, providing conditions for solution existence, uniqueness, and convergence in incomplete markets.

Contribution

It introduces a stability analysis framework for the Epstein-Zin problem under market perturbations, establishing conditions for solution stability and convergence.

Findings

Convexity of the problem domain is guaranteed under certain risk aversion and elasticity parameters.

Solutions such as consumption streams and utility processes converge as market perturbations diminish.

The paper provides a rigorous proof of solution existence and uniqueness in perturbed incomplete markets.

Abstract

We investigate the stability of the Epstein-Zin problem with respect to small distortions in the dynamics of the traded securities. We work in incomplete market model settings, where our parametrization of perturbations allows for joint distortions in returns and volatility of the risky assets and the interest rate. Considering empirically the most relevant specifications of risk aversion and elasticity of intertemporal substitution, we provide a condition that guarantees the convexity of the domain of the underlying problem and results in the existence and uniqueness of a solution to it. Then, we prove the convergence of the optimal consumption streams, the associated wealth processes, the indirect utility processes, and the value functions in the limit when the model perturbations vanish.