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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: rieder@mathematik.uni-ulm.de
∗∗Postal address: Institute for Mathematical Stochastics, University of Hannover, D-30167 Hannover, Germany. Email address: baeuerle@stochastik.uni-hannover.de
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Abstract

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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 

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