Before revision:  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.
After revision:  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 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.