When Bayesian learning meets stochastic optimal control : Optimal portfolio choice under drift uncertainty
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