The Annals of Applied Probability

Optimal consumption under habit formation in markets with transaction costs and random endowments

Xiang Yu

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This paper studies the optimal consumption via the habit formation preference in markets with transaction costs and unbounded random endowments. To model the proportional transaction costs, we adopt the Kabanov’s multi-asset framework with a cash account. At the terminal time $T$, the investor can receive unbounded random endowments for which we propose a new definition of acceptable portfolios based on the strictly consistent price system (SCPS). We prove a type of super-hedging theorem using the acceptable portfolios which enables us to obtain the consumption budget constraint condition under market frictions. Following similar ideas in [Ann. Appl. Probab. 25 (2015) 1383–1419] with the path dependence reduction and the embedding approach, we obtain the existence and uniqueness of the optimal consumption using some auxiliary processes and the duality analysis. As an application of the duality theory, the market isomorphism with special discounting factors is also discussed in the sense that the original optimal consumption with habit formation is equivalent to the standard optimal consumption problem without the habits impact, however, in a modified isomorphic market model.

Article information

Ann. Appl. Probab., Volume 27, Number 2 (2017), 960-1002.

Received: August 2014
Revised: December 2015
First available in Project Euclid: 26 May 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 91G10: Portfolio theory 91B42: Consumer behavior, demand theory
Secondary: 60G44: Martingales with continuous parameter 49K30: Optimal solutions belonging to restricted classes

Proportional transaction costs unbounded random endowments acceptable portfolios consumption budget constraint consumption habit formation convex duality market isomorphism


Yu, Xiang. Optimal consumption under habit formation in markets with transaction costs and random endowments. Ann. Appl. Probab. 27 (2017), no. 2, 960--1002. doi:10.1214/16-AAP1222.

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