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Optimal consumption of multiple goods in incomplete markets

Part of: Stochastic systems and control Mathematical finance

Published online by Cambridge University Press:  16 November 2018

Oleksii Mostovyi
Oleksii Mostovyi*
Affiliation:
University of Connecticut
*
* Postal address: Department of Mathematics, University of Connecticut, 341 Mansfield Road U1009, Storrs, CT 06269-1009, USA. Email address: oleksii.mostovyi@uconn.edu
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Abstract

We consider the problem of optimal consumption of multiple goods in incomplete semimartingale markets. We formulate the dual problem and identify conditions that allow for the existence and uniqueness of the solution, and provide a characterization of the optimal consumption strategy in terms of the dual optimizer. We illustrate our results with examples in both complete and incomplete models. In particular, we construct closed-form solutions in some incomplete models.

Keywords

Optimal consumptionmultiple goodsutility maximizationno unbounded profit with bounded riskarbitrage of the first kindlocal martingale deflatorduality theorysemimartingaleincomplete marketoptimal investment

MSC classification

Primary: 91G10: Portfolio theory 93E20: Optimal stochastic control
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 810 - 822
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Aliprantis, C. D. and Border, K. C. (2006). Infinite Dimensional Analysis, 3rd edn. Springer, Berlin.Google Scholar
[2]Breeden, D. (1979). An intertemporal asset pricing model with stochastic consumption and investment opportunities. J. Financial Econom. 7, 265–296.Google Scholar
[3]Chau, H. N., Cosso, A., Fontana, C. and Mostovyi, O. (2017). Optimal investment with intermediate consumption under no unbounded profit with bounded risk. J. Appl. Prob. 54, 710–719.Google Scholar
[4]Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520.Google Scholar
[5]Fischer, S. (1975). The demand for index bonds. J. Political Economy 83, 509–534.Google Scholar
[6]Goll, T. and Kallsen, J. (2000). Optimal portfolios for logarithmic utility. Stoch. Process. Appl. 89, 31–48.Google Scholar
[7]Hiriart-Urruty, J.-B. and Lemaréchal, C. (2001). Fundamentals of Convex Analysis. Springer, Berlin.Google Scholar
[8]Inada, K.-I. (1963). On a two-sector model of economic growth: comments and a generalization. Rev. Econom. Studies 30, 119–127.Google Scholar
[9]Kabanov, Y., Kardaras, C. and Song, S. (2016). No arbitrage of the first kind and local martingale numéraires. Finance Stoch. 20, 1097–1108.Google Scholar
[10]Karatzas, I. and Kardaras, K. (2007). The numéraire portfolio in semimartingale financial models. Finance Stoch. 11, 447–493.Google Scholar
[11]Karatzas, I. and Shreve, S. E. (1998). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
[12]Kardaras, C. (2010). Finitely additive probabilities and the fundamental theorem of asset pricing. In Contemporary Quantitative Finance, Springer, Berlin, pp. 19–34.Google Scholar
[13]Kardaras, C. (2012). Market viability via absence of arbitrage of the first kind. Finance Stoch. 16, 651–667.Google Scholar
[14]Kardaras, C. (2014). A time before which insiders would not undertake risk. In Inspired by Finance, Springer, Cham, pp. 349–362.Google Scholar
[15]Kim, T. S. and Omberg, E. (1996). Dynamic nonmyopic portfolio behavior. Rev. Financial Studies 9, 141–161.Google Scholar
[16]Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Prob. 9, 904–950.Google Scholar
[17]Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econom. Statist. 51, 247–257.Google Scholar
[18]Merton, R. C. (1992). Continuous-Time Finance. Blackwell, Malden, MA.Google Scholar
[19]Mostovyi, O. (2015). Necessary and sufficient conditions in the problem of optimal investment with intermediate consumption. Finance Stoch. 19, 135–159.Google Scholar
[20]Mostovyi, O. (2017). Optimal investment with intermediate consumption and random endowment. Math. Finance 27, 96–114.Google Scholar
[21]Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press.Google Scholar
[22]Siorpaes, P. (2015). Optimal investment and price dependence in a semi-static market. Finance Stoch. 19, 161–187.Google Scholar
[23]Siorpaes, P. (2016). Do arbitrage-free prices come from utility maximization? Math. Finance 26, 602–616.Google Scholar
[24]Takaoka, K. and Schweizer, M. (2014). A note on the condition of no unbounded profit with bounded risk. Finance Stoch. 18, 393–405.Google Scholar
[25]Tehranchi, M. (2004). Explicit solutions of some utility maximization problems in incomplete markets. Stoch. Process. Appl. 114, 109–125.Google Scholar