Journal of Applied Probability

Portfolio optimization with unobservable Markov-modulated drift process

Ulrich Rieder and Nicole Bäuerle

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Abstract

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.

Article information

Source
J. Appl. Probab. Volume 42, Number 2 (2005), 362-378.

Dates
First available in Project Euclid: 14 June 2005

Permanent link to this document
http://projecteuclid.org/euclid.jap/1118777176

Digital Object Identifier
doi:10.1239/jap/1118777176

Mathematical Reviews number (MathSciNet)
MR2145482

Zentralblatt MATH identifier
1138.93428

Subjects
Primary: 93E20: Optimal stochastic control

Keywords
Portfolio optimization Markov-modulated drift Hamilton-Jacobi-Bellman equation optimal investment strategy Bayesian control stochastic ordering

Citation

Rieder, Ulrich; Bäuerle, Nicole. Portfolio optimization with unobservable Markov-modulated drift process. J. Appl. Probab. 42 (2005), no. 2, 362--378. doi:10.1239/jap/1118777176. http://projecteuclid.org/euclid.jap/1118777176.


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References

  • Bäuerle, N. and Rieder, U. (2004). Portfolio optimization with Markov-modulated stock prices and interest rates. IEEE Trans. Automatic Control 49, 442--447.
  • Elliott, R. J., Aggoun, L. and Moore, J. B. (1994). Hidden Markov Models: Estimation and Control. Springer, New York.
  • 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.
  • Karatzas, I. and Zhao, X. (2001). Bayesian adaptive portfolio optimization. In Option Pricing, Interest Rates and Risk Management, Cambridge University Press, pp. 632--669.
  • Kloeden, P. E. and Platen, E. (1995). Numerical Solution of Stochastic Differential Equations. Springer, Berlin.
  • Kuwana, Y. (1991). Certainty equivalence and logarithmic utilities in consumption/investment problems. Math. Finance 5, 297--310.
  • Lakner, P. (1995). Utility maximization with partial information. Stoch. Process. Appl. 56, 247--273.
  • Lakner, P. (1998). Optimal trading strategy for an investor: the case of partial information. Stoch. Process. Appl. 76, 77--97.
  • 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.)
  • Pham, H. (2002). Smooth solutions to optimal investment models with stochastic volatilities and portfolio constraints. Appl. Math. Optimization 46, 55--78.
  • Rishel, R. (1999). Optimal portfolio management with partial observation and power utility function. In Stochastic Analysis, Control, Optimization and Applications, eds W. McEneaney, G. Yin and Q. Zhang, Birkhäuser, Boston, MA, pp. 605--620.
  • 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.
  • Zariphopoulou, T. (2001). A solution approach to valuation with unhedgeable risks. Finance Stoch. 5, 61--82.