Epstein-Zin Utility Maximization on a Random Horizon

Abstract

This paper solves the consumption-investment problem under Epstein-Zin preferences on a random horizon. In an incomplete market, we take the random horizon to be a stopping time adapted to the market filtration, generated by all observable, but not necessarily tradable, state processes. Contrary to prior studies, we do not impose any fixed upper bound for the random horizon, allowing for truly unbounded ones. Focusing on the empirically relevant case where the risk aversion and the elasticity of intertemporal substitution are both larger than one, we characterize the optimal consumption and investment strategies using backward stochastic differential equations with superlinear growth on unbounded random horizons. This characterization, compared with the classical fixed-horizon result, involves an additional stochastic process that serves to capture the randomness of the horizon. As demonstrated in two concrete examples, changing from a fixed horizon to a random one drastically alters the optimal strategies.

References in corpus (4)

• Comparison between the probability distribution of returns in the Heston model and empirical data for stock indexes
• The escape problem under stochastic volatility: the Heston model
• Consumption, Investment, and Healthcare with Aging
• Mortality and Healthcare: a Stochastic Control Analysis under Epstein-Zin Preferences

Cited by in corpus (1)

• Consumption-investment optimization with Epstein-Zin utility in unbounded non-Markovian markets