This paper investigates optimal portfolio strategies in a market with partial information on the drift. The drift is modelled as a function of a continuous-time Markov chain with finitely many states which is not directly observable. Information on the drift is obtained from the observation of stock prices. Moreover, expert opinions in the form of signals at random discrete time points are included in the analysis. We derive the filtering equation for the return process and incorporate the

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... r into the state variables of the optimization problem. This problem is studied with dynamic programming methods. In particular, we propose a policy improvement method to obtain computable approximations of the optimal strategy. Numerical results are presented at the end. drifts tend to fluctuate randomly over time; second, even if drifts were constant, a long time series is needed to estimate this parameter with a reasonable degree of precision as drift effects are usually dominated by volatility. For these reasons practitioners rely mostly on external sources of information such as news, company reports or ratings and on their own intuitive views when determining an estimate for the future growth rate of an asset; these outside sources of information are labelled expert opinions in this paper. The popular Black-Litterman model (see [2, 19] where subjective views are used to update equilibrium-implied returns in a Bayesian way is a typical example for the use of expert opinions in static (one-period) models. However, to the best of our knowledge expected utility maximization in dynamic portfolio optimization models with expert opinions has so far not been studied. In the present paper we set out to do exactly that. We consider a hidden Markov model (HMM) where asset prices follow a diffusion process whose drift is driven by an unobservable finite-state Markov chain Y . Information on the hidden chain is of mixed type. First, investors observe stock prices. Moreover, and this is the novel feature of this paper, expert opinions are included in the analysis as a second source of information. Mathematically, expert opinions are represented by a marked point process with jump-size distribution depending on the current state of Y . Standard filtering results for HMMs and Bayesian updating are used to derive a finitedimensional filter for the state of the hidden Markov chain. This allows us to reduce the portfolio optimization problem to a problem under complete information where the new state variables are the filter distribution and the wealth of the investor. In this model the market is incomplete, as the investor filtration is partly generated by the non-tradable marked point process that models the expert opinions. This makes the application of duality methods and of the martingale approach to portfolio optimization relatively involved. Hence we resort to dynamic programming and work with the associated Hamilton-Jacobi-Bellman (HJB) equation instead. We consider the case of logarithmic and power utilities. In the latter case the HJB equation can be simplified by a change of measure and we end up with a quasi-linear integrodifferential equation. Finally we propose a policy improvement method to obtain an approximation of the optimal strategy. Portfolio optimization under partial information on the drift has been studied extensively over the last years. There are two popular model classes for the drift, linear Gaussian dynamics and HMMs. For Gaussian dynamics explicit solutions for the problem of optimizing the expected utility of terminal wealth are provided for example in Lakner [11], Brendle [4], Danilova et al. [5], where the last paper focuses on additional insider information. Utility maximization for a HMM model is investigated for example in Rieder and Bäuerle [16], Sass and Haussmann [17], Sass and Wunderlich [18] and Gabih et al. [10]. These approaches are generalized in Björk et al. [1]. In the present paper we follow Rieder and Bäuerle [16] for the setup of the HJB equation in a model with an unobservable drift modelled by a finite-state Markov chain. Moreover, we were inspired by the change of measure technique used among others by Nagai and Runggaldier [14] and Davis and Lleo [6].