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# Utility maximization with addictive consumption habit formation in incomplete semimartingale markets

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• Xiang Yu

## Abstract

This paper studies the continuous time utility maximization problem on consumption with addictive habit formation in incomplete semimartingale markets. Introducing the set of auxiliary state processes and the modified dual space, we embed our original problem into a time-separable utility maximization problem with a shadow random endowment on the product space $\mathbb{L}_+^0(\Omega\times [0,T],\mathcal{O},\overline{\mathbb{P}})$. Existence and uniqueness of the optimal solution are established using convex duality approach, where the primal value function is defined on two variables, that is, the initial wealth and the initial standard of living. We also provide sufficient conditions on the stochastic discounting processes and on the utility function for the well-posedness of the original optimization problem. Under the same assumptions, classical proofs in the approach of convex duality analysis can be modified when the auxiliary dual process is not necessarily integrable.

## Suggested Citation

• Xiang Yu, 2011. "Utility maximization with addictive consumption habit formation in incomplete semimartingale markets," Papers 1112.2940, arXiv.org, revised May 2015.
• Handle: RePEc:arx:papers:1112.2940
as

File URL: http://arxiv.org/pdf/1112.2940

## References listed on IDEAS

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1. Mehra, Rajnish & Prescott, Edward C., 1985. "The equity premium: A puzzle," Journal of Monetary Economics, Elsevier, vol. 15(2), pages 145-161, March.
2. Jerome B. Detemple & Fernando Zapatero, 1992. "Optimal Consumption-Portfolio Policies With Habit Formation," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 251-274.
3. Kramkov, D.O., 1994. "Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets," Discussion Paper Serie B 294, University of Bonn, Germany.
4. Julien Hugonnier & Dmitry Kramkov, 2004. "Optimal investment with random endowments in incomplete markets," Papers math/0405293, arXiv.org.
5. Gordan Zitkovic, 2005. "Utility Maximization with a Stochastic Clock and an Unbounded Random Endowment," Papers math/0503516, arXiv.org.
6. Detemple, Jerome B & Zapatero, Fernando, 1991. "Asset Prices in an Exchange Economy with Habit Formation," Econometrica, Econometric Society, vol. 59(6), pages 1633-1657, November.
7. Frank Riedel & Peter Bank, 2001. "Existence and structure of stochastic equilibria with intertemporal substitution," Finance and Stochastics, Springer, vol. 5(4), pages 487-509.
8. Detemple, Jerome B. & Karatzas, Ioannis, 2003. "Non-addictive habits: optimal consumption-portfolio policies," Journal of Economic Theory, Elsevier, vol. 113(2), pages 265-285, December.
9. Mark Schroder & Costis Skiadas, 2002. "An Isomorphism Between Asset Pricing Models With and Without Linear Habit Formation," Review of Financial Studies, Society for Financial Studies, vol. 15(4), pages 1189-1221.
10. (**), 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.
11. Bruno Bouchard & Huyên Pham, 2004. "Wealth-path dependent utility maximization in incomplete markets," Finance and Stochastics, Springer, vol. 8(4), pages 579-603, November.
12. Roman Muraviev, 2011. "Additive habit formation: Consumption in incomplete markets with random endowments," Papers 1106.2980, arXiv.org, revised Sep 2011.
13. repec:dau:papers:123456789/1803 is not listed on IDEAS
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## Citations

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Cited by:

1. Xiang Yu, 2014. "Optimal Consumption under Habit Formation In Markets with Transaction Costs and Random Endowments," Papers 1408.1382, arXiv.org, revised Jul 2016.
2. repec:eee:mateco:v:70:y:2017:i:c:p:74-89 is not listed on IDEAS
3. Fortin, Ines & Hlouskova, Jaroslava & Tsigaris, Panagiotis, 2016. "The Consumption-Investment Decision of a Prospect Theory Household," Economics Series 322, Institute for Advanced Studies.

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