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Duality for optimal consumption with randomly terminating income

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  • Ashley Davey
  • Michael Monoyios
  • Harry Zheng

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.

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  • Ashley Davey & Michael Monoyios & Harry Zheng, 2020. "Duality for optimal consumption with randomly terminating income," Papers 2011.00732, arXiv.org, revised May 2021.
    • Handle: RePEc:arx:papers:2011.00732
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    References listed on IDEAS

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    1. Julien Hugonnier & Dmitry Kramkov, 2004. "Optimal investment with random endowments in incomplete markets," Papers math/0405293, arXiv.org.
    2. Michael Monoyios, 2020. "Duality for optimal consumption under no unbounded profit with bounded risk," Papers 2006.04687, arXiv.org, revised Dec 2021.
    3. Philipp Grohs & Fabian Hornung & Arnulf Jentzen & Philippe von Wurstemberger, 2018. "A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations," Papers 1809.02362, arXiv.org, revised Jan 2023.
    4. Darrell Duffie & Thaleia Zariphopoulou, 1993. "Optimal Investment With Undiversifiable Income Risk," Mathematical Finance, Wiley Blackwell, vol. 3(2), pages 135-148, April.
    5. He, Hua & Pages, Henri F, 1993. "Labor Income, Borrowing Constraints, and Equilibrium Asset Prices," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 3(4), pages 663-696, October.
    6. Nicole El Karoui & Monique Jeanblanc-Picqué, 1998. "Optimization of consumption with labor income," Finance and Stochastics, Springer, vol. 2(4), pages 409-440.
    7. Ioannis Karatzas & Constantinos Kardaras, 2007. "The numéraire portfolio in semimartingale financial models," Finance and Stochastics, Springer, vol. 11(4), pages 447-493, October.
    8. Föllmer, Hans & Kramkov, D. O., 1997. "Optional decompositions under constraints," SFB 373 Discussion Papers 1997,31, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    9. Oleksii Mostovyi, 2015. "Necessary and sufficient conditions in the problem of optimal investment with intermediate consumption," Finance and Stochastics, Springer, vol. 19(1), pages 135-159, January.
    10. Bian, Baojun & Zheng, Harry, 2015. "Turnpike property and convergence rate for an investment model with general utility functions," Journal of Economic Dynamics and Control, Elsevier, vol. 51(C), pages 28-49.
    11. (**), Hui Wang & Jaksa Cvitanic & (*), Walter Schachermayer, 2001. "Utility maximization in incomplete markets with random endowment," Finance and Stochastics, Springer, vol. 5(2), pages 259-272.
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