Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-09T01:34:33.501Z Has data issue: false hasContentIssue false
  • English
  • Français

Portfolio optimization with unobservable Markov-modulated drift process

Part of: Stochastic systems and control

Published online by Cambridge University Press:  14 July 2016

Ulrich Rieder  and
Nicole Bäuerle
Ulrich Rieder*
Affiliation:
University of Ulm
Nicole Bäuerle*
Affiliation:
University of Hannover
*
∗Postal address: Department of Optimization and Operations Research, University of Ulm, D-89069 Ulm, Germany. Email address: [email protected]
∗∗Postal address: Institute for Mathematical Stochastics, University of Hannover, D-30167 Hannover, Germany. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study portfolio optimization problems in which the drift rate of the stock is Markov modulated and the driving factors cannot be observed by the investor. Using results from filter theory, we reduce this problem to one with complete observation. In the cases of logarithmic and power utility, we solve the problem explicitly with the help of stochastic control methods. It turns out that the value function is a classical solution of the corresponding Hamilton-Jacobi-Bellman equation. As a special case, we investigate the so-called Bayesian case, i.e. where the drift rate is unknown but does not change over time. In this case, we prove a number of interesting properties of the optimal portfolio strategy. In particular, using the likelihood-ratio ordering, we can compare the optimal investment in the case of observable drift rate to that in the case of unobservable drift rate. Thus, we also obtain the sign of the drift risk.

Keywords

Portfolio optimizationMarkov-modulated driftHamilton-Jacobi-Bellman equationoptimal investment strategyBayesian controlstochastic ordering

MSC classification

Primary: 93E20: Optimal stochastic control
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 362 - 378
Copyright
© Applied Probability Trust 2005 

References

Bäuerle, N. and Rieder, U. (2004). Portfolio optimization with Markov-modulated stock prices and interest rates. IEEE Trans. Automatic Control 49, 442–447.CrossRefGoogle Scholar
Elliott, R. J., Aggoun, L. and Moore, J. B. (1994). Hidden Markov Models: Estimation and Control. Springer, New York.Google Scholar
Haussmann, U. G. and Sass, J. (2004). Optimal terminal wealth under partial information for HMM stock returns. In Mathematics of Finance (Contemp. Math. 351), AMS, Providence, RI, pp. 171–185.CrossRefGoogle Scholar
Karatzas, I. and Zhao, X. (2001). Bayesian adaptive portfolio optimization. In Option Pricing, Interest Rates and Risk Management, Cambridge University Press, pp. 632–669.CrossRefGoogle Scholar
Kloeden, P. E. and Platen, E. (1995). Numerical Solution of Stochastic Differential Equations. Springer, Berlin.Google Scholar
Kuwana, Y. (1991). Certainty equivalence and logarithmic utilities in consumption/investment problems. Math. Finance 5, 297–310.CrossRefGoogle Scholar
Lakner, P. (1995). Utility maximization with partial information. Stoch. Process. Appl. 56, 247–273.CrossRefGoogle Scholar
Lakner, P. (1998). Optimal trading strategy for an investor: the case of partial information. Stoch. Process. Appl. 76, 77–97.CrossRefGoogle Scholar
Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3, 373–413. (Correction: 6 (1973), 213–214.)CrossRefGoogle Scholar
Pham, H. (2002). Smooth solutions to optimal investment models with stochastic volatilities and portfolio constraints. Appl. Math. Optimization 46, 55–78.CrossRefGoogle Scholar
Rishel, R. (1999). Optimal portfolio management with partial observation and power utility function. In Stochastic Analysis, Control, Optimization and Applications, eds McEneaney, W., Yin, G. and Zhang, Q., Birkhäuser, Boston, MA, pp. 605–620.CrossRefGoogle Scholar
Sass, J. and Haussmann, U. G. (2004). Optimizing the terminal wealth under partial information: the drift process as a continuous time Markov chain. Finance Stoch. 8, 553–577.CrossRefGoogle Scholar
Zariphopoulou, T. (2001). A solution approach to valuation with unhedgeable risks. Finance Stoch. 5, 61–82.CrossRefGoogle Scholar