We shall present several models addressing optimal portfolio
choice and optimal portfolio transition issues, in which the expected
returns of risky assets are unknown. Our approach is based on a
coupling between Bayesian learning and dynamic programming techniques.
It permits to recover the well-known results of Karatzas and Zhao in
the case of conjugate (Gaussian) priors for the drift distribution,
but also to go beyond the no-friction case, when martingale methods
are no longer available. In particular, we address optimal portfolio
choice in a framework à la Almgren-Chriss and we build therefore a
model in which the agent takes into account in his/her allocation
decision process both the liquidity of assets and the uncertainty with
respect to their expected returns. We also address optimal portfolio
liquidation and optimal portfolio transition problems.

Keywords : Bayesian learning, Hamilton-Jacobi-Bellman, duality, PDE,
portfolio optimization