## Journal of Applied Probability

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

### Portfolio optimization with unobservable Markov-modulated drift process

Ulrich Rieder and Nicole Bäuerle

#### 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.