Duality for optimal consumption with randomly terminating income

TL;DR

This paper develops a duality theory for infinite horizon optimal consumption with randomly terminating income, resolving a previous duality gap and characterizing optimal strategies under broad conditions.

Contribution

It establishes a rigorous duality framework for the problem, including the density of local martingale deflators and the characterization of optimal wealth processes.

Findings

Duality gap closed in the Black-Scholes market setting.

Optimal deflated wealth decays to zero at the optimum.

Dual variables include a dense set of local martingale deflators.

Abstract

We establish a rigorous duality theory, under No Unbounded Profit with Bounded Risk, for an infinite horizon problem of optimal consumption in the presence of an income stream that can terminate randomly at an exponentially distributed time, independent of the asset prices. We thus close a duality gap encountered by Vellekoop and Davis in a version of this problem in a Black-Scholes market. Many of the classical tenets of duality theory hold, with the notable exception that marginal utility at zero initial wealth is finite. We use as dual variables a class of supermartingale deflators such that deflated wealth plus cumulative deflated consumption in excess of income is a supermartingale. We show that the space of discounted local martingale deflators is dense in our dual domain, so that the dual problem can also be expressed as an infimum over the discounted local martingale deflators. We characterise the optimal wealth process, showing that optimal deflated wealth is a potential decaying to zero, while deflated wealth plus cumulative deflated consumption over income is a uniformly integrable martingale at the optimum. We apply the analysis to the Vellekoop and Davis example and give a numerical solution.