## The Annals of Applied Probability

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

Xiang Yu

#### Abstract

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

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

Dates
Revised: December 2015
First available in Project Euclid: 26 May 2017

https://projecteuclid.org/euclid.aoap/1495764372

Digital Object Identifier
doi:10.1214/16-AAP1222

Mathematical Reviews number (MathSciNet)
MR3655859

Zentralblatt MATH identifier
1368.91168

#### Citation

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. https://projecteuclid.org/euclid.aoap/1495764372

#### References

• [1] Benedetti, G. and Campi, L. (2012). Multivariate utility maximization with proportional transaction costs and random endowment. SIAM J. Control Optim. 50 1283–1308.
• [2] Campbell, J. Y. and Cochrane, J. H. (1999). By force of habit: A consumption-based explanation of aggregate stock market behavior. J. Polit. Econ. 107 205–251.
• [3] Campi, L. and Schachermayer, W. (2006). A super-replication theorem in Kabanov’s model of transaction costs. Finance Stoch. 10 579–596.
• [4] Constantinides, G. M. (1988). Habit formation: A resolution of the equity premium puzzle. Working paper, Center for Research in Security Prices, Graduate School of Business, Univ. Chicago, Chicago, IL.
• [5] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463–520.
• [6] Delbaen, F. and Schachermayer, W. (1997). The Banach space of workable contingent claims in arbitrage theory. Ann. Inst. Henri Poincaré Probab. Stat. 33 113–144.
• [7] Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 215–250.
• [8] Detemple, J. B. and Zapatero, F. (1991). Asset prices in an exchange economy with habit formation. Econometrica 59 1633–1657.
• [9] Detemple, J. B. and Zapatero, F. (1992). Optimal consumption-portfolio policies with habit formation. Math. Finance 2 251–274.
• [10] Englezos, N. and Karatzas, I. (2009). Utility maximization with habit formation: Dynamic programming and stochastic PDEs. SIAM J. Control Optim. 48 481–520.
• [11] Fuhrer, J. C. (2000). Habit formation in consumption and its implications for monetary-policy models. Am. Econ. Rev. 90 367–390.
• [12] Graves, L. M. (1946). The Theory of Functions of Real Variables. McGraw-Hill, New York.
• [13] Guasoni, P., Lépinette, E. and Rásonyi, M. (2012). The fundamental theorem of asset pricing under transaction costs. Finance Stoch. 16 741–777.
• [14] Guasoni, P., Rásonyi, M. and Schachermayer, W. (2010). The fundamental theorem of asset pricing for continuous processes under small transaction costs. Ann. Finance 6 157–191.
• [15] Hicks, J. (1965). Capital and Growth. Oxford Univ. Press, New York.
• [16] Hugonnier, J. and Kramkov, D. (2004). Optimal investment with random endowments in incomplete markets. Ann. Appl. Probab. 14 845–864.
• [17] Jacka, S. and Berkaoui, A. (2007). On the density of properly maximal claims in financial markets with transaction costs. Ann. Appl. Probab. 17 716–740.
• [18] Kabanov, Y. and Safarian, M. (2009). Markets with Transaction Costs: Mathematical Theory. Springer, Berlin.
• [19] Kabanov, Y. M. and Last, G. (2002). Hedging under transaction costs in currency markets: A continuous-time model. Math. Finance 12 63–70.
• [20] Kabanov, Y. M. and Stricker, C. (2002). Hedging of contingent claims under transaction costs. In Advances in Finance and Stochastics 125–136. Springer, Berlin.
• [21] Kardaras, C. and Platen, E. (2011). On the semimartingale property of discounted asset-price processes. Stochastic Process. Appl. 121 2678–2691.
• [22] Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9 904–950.
• [23] Kramkov, D. and Sîrbu, M. (2006). Sensitivity analysis of utility-based prices and risk-tolerance wealth processes. Ann. Appl. Probab. 16 2140–2194.
• [24] Kramkov, D. O. (1996). Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Related Fields 105 459–479.
• [25] Mehra, R. and Prescott, E. C. (1985). The equity premium: A puzzle. J. Monet. Econ. 15 145–161.
• [26] Muraviev, R. (2011). Additive habit formation: Consumption in incomplete markets with random endowments. Math. Financ. Econ. 5 67–99.
• [27] Rásonyi, M. (2003). A remark on the superhedging theorem under transaction costs. In Séminaire de Probabilités XXXVII. Lecture Notes in Math. 1832 394–398. Springer, Berlin.
• [28] Ryder, H. E. and Heal, G. M. (1973). Optimal growth with intertemporally dependent preferences. Rev. Econ. Stud. 40 1–33.
• [29] Schachermayer, W. (2004). The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Finance 14 19–48.
• [30] Schachermayer, W. (2004). Portfolio Optimization in Incomplete Financial Markets. Scuola Normale Superiore, Classe di Scienze, Pisa.
• [31] Schroder, M. and Skiadas, C. (2002). An isomorphism between asset pricing models with and without linear habit formation. Rev. Financ. Stud. 15 1189–1221.
• [32] Yu, X. (2015). Utility maximization with addictive consumption habit formation in incomplete semimartingale markets. Ann. Appl. Probab. 25 1383–1419.
• [33] Žitković, G. (2005). Utility maximization with a stochastic clock and an unbounded random endowment. Ann. Appl. Probab. 15 748–777.