  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.
