  We study the Markowitz portfolio selection problem with unknown drift vector
in the multidimensional framework. The prior belief on the uncertain expected
rate of return is modeled by an arbitrary probability law, and a Bayesian
approach from filtering theory is used to learn the posterior distribution
about the drift given the observed market data of the assets. The Bayesian
Markowitz problem is then embedded into an auxiliary standard control problem
that we characterize by a dynamic programming method and prove the existence
and uniqueness of a smooth solution to the related semi-linear partial
differential equation (PDE). The optimal Markowitz portfolio strategy is
explicitly computed in the case of a Gaussian prior distribution. Finally, we
measure the quantitative impact of learning, updating the strategy from
observed data, compared to non-learning, using a constant drift in an uncertain
context, and analyze the sensitivity of the value of information w.r.t. various
relevant parameters of our model.
