Duality for optimal consumption with randomly terminating income

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 in the Davis‐Vellekoop example 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 characterize 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 Davis‐Vellekoop example and give a numerical solution.

Optimal Investment, Heterogeneous Consumption, and Best Time for Retirement

“…1. The wealth process is normally assumed to be nonnegative for investment consumption problem when there is no income, but can be negative if there is a deterministic income which may offset the negative wealth, see Davey et al (2021) for discussions on that point. In our model the wealth process can go negative range within a certain boundary determined by considering the current value of net-income amount being able to earn for a whole lifetime.…”