Convex duality for stochastic differential utility

TL;DR

This paper develops a duality framework for continuous-time consumption and investment problems with stochastic differential utility, enabling the identification of optimal strategies and market completions without restrictive assumptions.

Contribution

It establishes duality for Epstein-Zin utility in incomplete markets, providing a novel method to determine optimal strategies and market completions.

Findings

Duality between primal and dual problems is established.

Optimal strategies are identified without restrictive assumptions.

The dual minimizer is interpreted as the 'least favorable' market completion.

Abstract

This paper introduces a dual problem to study a continuous-time consumption and investment problem with incomplete markets and stochastic differential utility. For Epstein-Zin utility, duality between the primal and dual problems is established. Consequently the optimal strategy of the consumption and investment problem is identified without assuming several technical conditions on market model, utility specification, and agent's admissible strategy. Meanwhile the minimizer of the dual problem is identified as the utility gradient of the primal value and is economically interpreted as the "least favorable" completion of the market.