We establish a rigorous duality theory for a finite 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 an infinite horizon version of this problem . Nearly all the classical tenets of duality theory are found to hold, with the notable exception that the marginal utility of initial wealth at zero is finite. The intuition is that the agent will receive some income , no matter how early it terminates, so is not infinitely penalised for having zero initial capital. We then solve the problem numerically, with an additional terminal wealth objective, using deep learning. We transform the problem with randomly terminating income into one that no longer depends on the jump component but has an additional inter-temporal wealth objective. We then numerically solve the second order backward stochastic differential equations (2BSDEs), in both the primal and dual dimensions, to find the optimal control and tight lower and upper bounds for the value function .
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 .
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the marginal utility of initial wealth at zero
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The intuition is that the agent will receive some income , no matter how early it terminates, so is not infinitely penalised for having zero initial capital. We then solve the problem numerically, with an additional terminal wealth objective, using deep learning. We transform the problem with randomly terminating income into one that no longer depends on the jump component but has an additional inter-temporal wealth objective. We then numerically solve the second order backward stochastic differential equations (2BSDEs), in both the primal and dual dimensions, to find the optimal control and tight lower and upper bounds for the value function
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