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Optimal investment with intermediate consumption under no unbounded profit with bounded risk

Part of: Mathematical finance Stochastic systems and control

Published online by Cambridge University Press:  15 September 2017

Huy N. Chau ,
Andrea Cosso ,
Claudio Fontana  and
Oleksii Mostovyi
Huy N. Chau*
Affiliation:
Hungarian Academy of Sciences
Andrea Cosso*
Affiliation:
Politecnico di Milano
Claudio Fontana*
Affiliation:
Paris Diderot University
Oleksii Mostovyi*
Affiliation:
University of Connecticut
*
* Postal address: Alfred Renyi Institute of Mathematics, Hungarian Academy of Sciences, Realtanoda utca 13-15, H-1053 Budapest, Hungary. Email address: chau@renyi.hu
** Postal address: Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy. Email address: andrea.cosso@polimi.it
*** Postal address: Laboratoire de Probabilités et Modèles Aléatoires, Paris Diderot University, av. de France, 75205 Paris, France. Email address: fontana@math.univ-paris-diderot.fr
**** Postal address: Department of Mathematics, University of Connecticut, U1009, 341 Mansfield Road, Storrs, CT 06269-1009, USA. Email address: oleksii.mostovyi@uconn.edu
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Abstract

We consider the problem of optimal investment with intermediate consumption in a general semimartingale model of an incomplete market, with preferences being represented by a utility stochastic field. We show that the key conclusions of the utility maximization theory hold under the assumptions of no unbounded profit with bounded risk and of the finiteness of both primal and dual value functions.

Keywords

Utility maximizationarbitrage of the first kindlocal martingale deflatorduality theorysemimartingaleincomplete market

MSC classification

Primary: 91G10: Portfolio theory
Secondary: 93E20: Optimal stochastic control
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 710 - 719
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Aksamit, A., Choulli, T., Deng, J. and Jeanblanc, M. (2014). Non-arbitrage up to random horizon for semimartingale models. Preprint. Available at https://arxiv.org/abs/1310.1142. Google Scholar
[2] Ansel, J.-P. and Stricker, C. (1994). Couverture des actifs contingents et prix maximum. Ann. Inst. H. Poincaré Prob. Statist. 30, 303–315. Google Scholar
[3] Brannath, W. and Schachermayer, W. (1999). A bipolar theorem for L 0 +(Ω, F, P). In Séminaire de Probabilités, XXXIII (Lecture Notes Math. 1709). Springer, Berlin, pp. 349–354. CrossRefGoogle Scholar
[4] Choulli, T., Deng, J. and Ma, J. (2015). How non-arbitrage, viability and numéraire portfolio are related. Finance Stoch. 19, 719–741. CrossRefGoogle Scholar
[5] Christensen, M. M. and Larsen, K. (2007). No arbitrage and the growth optimal portfolio. Stoch. Anal. Appl. 25, 255–280. Google Scholar
[6] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520. Google Scholar
[7] Delbaen, F. and Schachermayer, W. (1995). Arbitrage possibilities in Bessel processes and their relations to local martingales. Prob. Theory Relat. Fields 102, 357–366. Google Scholar
[8] Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250. Google Scholar
[9] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential B. North-Holland, Amsterdam. Google Scholar
[10] Föllmer, H. and Kramkov, D. (1997). Optional decomposition under constraints. Prob. Theory Relat. Fields 109, 1–25. Google Scholar
[11] Fontana, C. and Runggaldier, W. (2013). Diffusion-based models for financial markets without martingale measures. In Risk Measures and Attitudes, Springer, London, pp. 45–81. Google Scholar
[12] Fontana, C., Øksendal, B. and Sulem, A. (2015). Market viability and martingale measures under partial information. Methodol. Comput. Appl. Prob. 17, 15–39. Google Scholar
[13] Harrison, J. M. and Kreps, D. M. (1979). Martingales and arbitrage in multiperiod securities markets. J. Econom. Theory 20, 381–408. Google Scholar
[14] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin. Google Scholar
[15] Karatzas, I. and Fernholz, R. (2009). Stochastic portfolio theory: an overview. In Handbook of Numerical Analysis, Vol. 15, North-Holland, Amsterdam, pp. 89–167. Google Scholar
[16] Karatzas, I. and Kardaras, C. (2007). The numéraire portfolio in semimartingale financial models. Finance Stoch. 11, 447–493. Google Scholar
[17] Karatzas, I. and Žitković, G. (2003). Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Prob. 31, 1821–1858. Google Scholar
[18] Karatzas, I., Lehoczky, J. P., Shreve, S. E. and Xu, G.-L. (1991). Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control Optimization 29, 702–730. Google Scholar
[19] Kardaras, C. (2010). Finitely additive probabilities and the fundamental theorem of asset pricing. In Contemporary Quantitative Finance, Springer, Berlin, pp. 19–34. Google Scholar
[20] Kardaras, C. (2012). Market viability via absence of arbitrage of the first kind. Finance Stoch. 16, 651–667. CrossRefGoogle Scholar
[21] Kardaras, C. (2013). On the closure in the Emery topology of semimartingale wealth-process sets. Ann. Appl. Prob. 23, 1355–1376. Google Scholar
[22] Kardaras, C. (2014). A time before which insiders would not undertake risk. In Inspired by Finance, Springer, Cham, pp. 349–362. Google Scholar
[23] Klein, I., Lépinette, E. and Perez-Ostafe, L. (2014). Asymptotic arbitrage with small transaction costs. Finance Stoch. 18, 917–939. Google Scholar
[24] 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
[25] Kramkov, D. and Schachermayer, W. (2003). Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Ann. Appl. Prob. 13, 1504–1516. Google Scholar
[26] Larsen, K. (2009). Continuity of utility-maximization with respect to preferences. Math. Finance 19, 237–250. Google Scholar
[27] Larsen, K. and Žitković, G. (2013). On utility maximization under convex portfolio constraints. Ann. Appl. Prob. 23, 665–692. CrossRefGoogle Scholar
[28] Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econom. Statist. 51, 247–257. Google Scholar
[29] Mostovyi, O. (2015). Necessary and sufficient conditions in the problem of optimal investment with intermediate consumption. Finance Stoch. 19, 135–159. Google Scholar
[30] Stricker, C. and Yan, J. A. (1998). Some remarks on the optional decomposition theorem. In Séminaire de Probabilités, XXXII (Lecture Notes Math. 1686), Springer, Berlin, pp. 56–66. Google Scholar
[31] 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
[32] Žitković, G. (2005). Utility maximization with a stochastic clock and an unbounded random endowment. Ann. Appl. Prob. 15, 748–777. CrossRefGoogle Scholar
[33] Žitković, G. (2010). Convex compactness and its applications. Math. Financial Econom. 3, 1–12. Google Scholar