Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T13:40:11.179Z Has data issue: false hasContentIssue false
  • English
  • Français

Anticipative portfolio optimization

Part of: Stochastic analysis Stochastic processes Stochastic systems and control

Published online by Cambridge University Press:  01 July 2016

Igor Pikovsky  and
Ioannis Karatzas
Igor Pikovsky*
Affiliation:
Carnegie Mellon University
Ioannis Karatzas*
Affiliation:
Columbia University
*
∗ Postal address: Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213, USA. [email protected].
∗∗ Postal address: Departments of Mathematics and Statistics, Columbia University, New York, NY 10027, USA. [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a classical stochastic control problem arising in financial economics: to maximize expected logarithmic utility from terminal wealth and/or consumption. The novel feature of our work is that the portfolio is allowed to anticipate the future, i.e. the terminal values of the prices, or of the driving Brownian motion, are known to the investor, either exactly or with some uncertainty. Results on the finiteness of the value of the control problem are obtained in various setups, using techniques from the so-called enlargement of filtrations. When the value of the problem is finite, we compute it explicitly and exhibit an optimal portfolio in closed form.

Keywords

OPTIMAL PORTFOLIOSSTOCHASTIC CONTROLSTOCHASTIC ANALYSISENLARGEMENT OF FILTRATIONSRELATIVE ENTROPY

MSC classification

Primary: 93E20: Optimal stochastic control 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G44: Martingales with continuous parameter
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 4 , December 1996 , pp. 1095 - 1122
Copyright
Copyright © Applied Probability Trust 1996 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported in part by the National Science Foundation under Grant NSF-DMS-93-19816.

References

[1] Al Hussaini, A. and Elliott, R. (1987) Enlarged filtrations for diffusions. Stoch. Proc. Appl. 24, 99–107.Google Scholar
[2] Back, K. (1992) Insider trading in continuous time. Rev. Finan. Stud. 5, 387–409.CrossRefGoogle Scholar
[3] Back, K. (1993) Time-varying liquidity trading, price pressure, and volatility. Preprint, Washington University, St. Louis.Google Scholar
[4] Bensoussan, A. (1984) On the theory of option pricing. Acta Appl. Math. 2, 139–158.Google Scholar
[5] Chaleyat-Maurel, M. and Jeulin, T. (1985) Grossisement Gaussien de la filtration Brownienne. In Grossisements de Filtrations: Exemples et Applications. ed. Jeulin, T. and Yor, M. Springer, Berlin pp. 59–109.Google Scholar
[6] Cvitanic, J. and Karatzas, I. (1992) Convex duality in constrained portfolio optimization. Ann. Appl. Prob. 2, 767–818.Google Scholar
[7] Föllmer, H. and Imkeller, P. (1993) Anticipation cancelled by a Girsanov transformation: A paradox on Wiener space. Ann. Inst. H. Poincaré Prob. Statist. 29, 569–586.Google Scholar
[8] Jacod, J. (1985) Grossisement initial, hypothèse (H'), et théorème de Girsanov. In Grossisements de Filtrations: Exemples et Applications. ed. Jeulin, T. and Yor, M. Springer, Berlin. pp. 15–35.Google Scholar
[9] Jeulin, J. (1980) Semi-Martingales et Groissisement d'une Filtration. (Lecture Notes in Mathematics 833.) Springer, Berlin.CrossRefGoogle Scholar
[10] Jeulin, T. and Yor, M. (1979) Inegalité de Hardy, semimartingales, et faux-amis. Séminaire de Probabilités 12. (Lecture Notes in Mathematics 721.) Springer, Berlin.Google Scholar
[11] Jeulin, T. and Yor, M. (1985) Grossisements de Filtrations: Exemples et Applications. (Lecture Notes in Mathematics 1118.) Springer, Berlin.Google Scholar
[12] Karatzas, I. (1989) Optimization problems in the theory of continuous trading. SIAM J. Control Optim. 27, 1221–1259.CrossRefGoogle Scholar
[13] Karatzas, I., Lehoczky, J. P. and Shreve, S. E. (1987) Optimal portfolio and consumption decisions for a ‘small investor’ on a finite horizon. SIAM J. Control Optim. 25, 1557–1586.Google Scholar
[14] Karatzas, I., Lehoczky, J. P., Shreve, S. E. and Xu, G. (1991) Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control Optim. 29, 702–730.Google Scholar
[15] Karatzas, I. and Shreve, S. E. (1991) Brownian Motion and Stochastic Calculus. 2nd edn. Springer, New York.Google Scholar
[16] Kyle, A. (1985) Continuous auctions and insider trading. Econometrica 53, 1315–1335.Google Scholar