An optimal investment problem with randomly terminating income

Vellekoop, Michel; Davis, Mark H. A.

doi:10.1109/cdc.2009.5400455

Cited by 6 publications

(32 citation statements)

References 12 publications

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“…Finally, we analyze the Vellekoop and Davis (2009) example, exploring the ramifications of duality, and numerically solving the problem. By integrating over the distribution of the random income termination time, we transform the pre‐income termination problem to a stochastic control problem with perpetual income in which utility is derived from consumption and inter‐temporal wealth, and numerically solve the HJB equation for the resulting problem.…”

Section: Introductionmentioning

confidence: 99%

“…The remainder of the paper is structured as follows. In Section 2 we formulate the primal problem, budget constraint and the dual problem and outline the Vellekoop and Davis (2009) example. In Section 3 we state the main duality theorem (Theorem 3.1) and also Theorem 3.2, that the infima over consumption deflators and discounted local martingale deflators coincide.…”

Section: Introductionmentioning

confidence: 99%

“…The proofs of these results are given in Section 5. In Section 6 we examine the Vellekoop and Davis (2009) example. Section 7 concludes.…”

Section: Introductionmentioning

confidence: 99%

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Duality for optimal consumption with randomly terminating income

Davey

Monoyios

Zheng

2021

Mathematical Finance

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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.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Duality for optimal consumption with randomly terminating income

Davey

Monoyios

Zheng

2021

Mathematical Finance

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“…In this paper we solve an optimal consumption problem in an otherwise complete financial market, in which incompleteness is generated by the presence of a randomly terminating income stream, which pays at a constant rate until some exponential random time, independent of the asset prices, at which time the income abruptly stops. We consider a finite horizon version of this problem, for which Vellekoop and Davis [35] considered the infinite horizon case in a Black-Scholes market.…”

Section: Introductionmentioning

confidence: 99%

“…Despite the apparent simplicity of the modification of the classical Merton problem with deterministic (or indeed no) income, the problem in [35] proved a remarkably intractable one to solve and understand. Vellekoop and Davis implemented a differential-equation based dual approach, made an ansatz that the dual control was deterministic, or at least did not depend on the termination time, and also used some differential equation heuristics to argue that the derivative of the value function at zero initial wealth was infinite.…”

Section: Introductionmentioning

confidence: 99%

Duality for optimal consumption with randomly terminating income

Davey,

Monoyios,

Zheng

2020

Preprint

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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 [35] 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.

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