Consumption investment optimization with Epstein-Zin utility in incomplete markets

TL;DR

This paper analyzes optimal consumption and investment strategies in incomplete markets with Epstein-Zin utility, addressing mathematical challenges and providing characterizations relevant for asset pricing puzzles.

Contribution

It characterizes optimal strategies using backward stochastic differential equations for Epstein-Zin utility with risk aversion and elasticity of substitution greater than one.

Findings

Optimal strategies derived via BSDEs

Superdifferential of indirect utility obtained

Addresses utility specification challenges in optimization

Abstract

In a market with stochastic investment opportunities, we study an optimal consumption investment problem for an agent with recursive utility of Epstein-Zin type. Focusing on the empirically relevant specification where both risk aversion and elasticity of intertemporal substitution are in excess of one, we characterize optimal consumption and investment strategies via backward stochastic differential equations. The supperdifferential of indirect utility is also obtained, meeting demands from applications in which Epstein-Zin utilities were used to resolve several asset pricing puzzles. The empirically relevant utility specification introduces difficulties to the optimization problem due to the fact that the Epstein-Zin aggregator is neither Lipschitz nor jointly concave in all its variables.