  This paper presents several models addressing optimal portfolio choice,
optimal portfolio liquidation, 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 that
leads to partial differential equations. It enables to recover the well-known
results of Karatzas and Zhao in a framework \`a la Merton, but also to deal
with cases where martingale methods are no longer available. In particular, we
address optimal portfolio choice, portfolio liquidation, and portfolio
transition problems in a framework \`a la Almgren-Chriss, and we build
therefore a model in which the agent takes into account in his decision process
both the liquidity of assets and the uncertainty with respect to their expected
return.
