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A note on long-term optimal portfolios under drawdown constraints

Part of: Miscellaneous topics in calculus of variations and optimal control Stochastic systems and control

Published online by Cambridge University Press:  01 July 2016

Jun Sekine
Jun Sekine*
Affiliation:
Kyoto University
*
∗ Postal address: Institute of Economic Research, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, 606-8501, Japan. Email address: sekine@kier.kyoto-u.ac.jp
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Abstract

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The maximization of the long-term growth rate of expected utility is considered under drawdown constraints. In a general situation, the value and the optimal strategy of the problem are related to those of another ‘standard’ risk-sensitive-type portfolio optimization problem. Furthermore, an upside-chance maximization problem of a large deviation probability is stated as a ‘dual’ optimization problem. As an example, a ‘linear-quadratic’ model is studied in detail: the conditions to ensure the solvabilities of the problems are discussed and explicit expressions for the solutions are presented.

Keywords

Drawdown constraintrisk constraintlong-term investmentrisk-sensitive controlSkorokhod equationalgebraic Riccati equation

MSC classification

Primary: 93E20: Optimal stochastic control 90C39: Dynamic programming 49N10: Linear-quadratic problems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 3 , September 2006 , pp. 673 - 692
Copyright
Copyright © Applied Probability Trust 2006 

References

Artzner, P. et al. (2004). Coherent multiperiod risk adjusted values and Bellman's principle. To appear in Ann. Operat. Res. Google Scholar
Bielecki, T. R. and Pliska, S. R. (2004). Risk sensitive intertemporal CAPM, with application to fixed-income management. IEEE Trans. Automatic Control 49, 420–432.Google Scholar
Brockett, R.W. (1970). Finite Dimensional Linear Systems. John Wiley. New York.Google Scholar
Cheridito, P., Delbaen, F. and Kupper, M. (2005). Coherent and convex monetary risk measures for unbounded càdlàg processes. Finance Stoch. 9, 369–387.Google Scholar
Cvitanić, J. and Karatzas, I. (1995). On portfolio optimization under ‘drawdown’ constraints. In Mathematical Finance (IMA Lecture Notes Math. Appl. 65), Springer, New York, pp. 35–45.Google Scholar
Fleming, W. H. and Sheu, S. J. (2000). Risk-sensitive control and an optimal investment model. Math. Finance 10, 197–213.Google Scholar
Fleming, W. H. and Sheu, S. J. (2002). Risk-sensitive control and an optimal investment model. II. Ann. Appl. Prob. 12, 730–767.Google Scholar
Grossman, S. J. and Zhou, Z. (1993). Optimal investment strategies for controlling drawdowns. Math. Finance 3, 241–276.Google Scholar
Hata, H. and Iida, Y. (2005). A risk-sensitive stochastic control approach to an optimal investment problem with partial information. To appear in Finance Stoch.Google Scholar
Hata, H. and Sekine, J. (2005). Solving long term optimal investment problems with Cox–Ingersoll–Ross interest rates. Adv. Math. Econom. 8, 231–255.Google Scholar
Inoue, A. and Nakano, Y. (2005). Optimal long term investment model with memory. To appear in Appl. Math. Optimization. Google Scholar
Kaise, H. and Sheu, S. J. (2004). Risk sensitive optimal investment: solutions of the dynamical programming equation. In Mathematics of Finance (Contemp. Math. 351), American Mathematical Society, Providence, RI, pp. 217–230.Google Scholar
Karatzas, I. and Shreve, S. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York.Google Scholar
Kuroda, K. and Nagai, H. (2002). Risk sensitive portfolio optimization on infinite time horizon. Stoch. Stoch. Reports 73, 309–331.Google Scholar
Lancaster, P. and Rodman, L. (1995). Algebraic Riccati Equations. Oxford Science Publications.Google Scholar
Nagai, H. (2003). Optimal strategies for risk-sensitive portfolio optimization problems for general factor models. SIAM J. Control Optimization 41, 1779–1800.CrossRefGoogle Scholar
Pham, H. (2003). A large deviations approach to optimal long term investment. Finance Stoch. 7, 169–195.Google Scholar
Pham, H. (2003). A risk-sensitive control dual approach to a large deviations control problem. Systems Control Lett. 49, 295–309.CrossRefGoogle Scholar
Sekine, J. (2006). On a large deviations control for a linear-quadratic model: the complete dual solution. To appear in Proceedings of the 4th JSIAM-SIMAI Meeting.Google Scholar
Willems, J. C. (1971). Least squares stationary optimal control and the algebraic Riccati equation. IEEE Trans. Automatic Control 16, 621–634.Google Scholar