This paper presents 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 \`a 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.
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