We investigate joint optimization on information acquisition and portfolio selection within a Bayesian adaptive framework. The investor dynamically controls the precision of a private signal and incurs costs while updating her belief about the unobservable asset drift. Controllable information acquisition fails the classical separation principle of stochastic filtering. We adopt functional modeling of control to address the consequential endogeneity issues, then solve our optimization problem through dynamic programming. When the unknown drift follows a Gaussian prior, the HJB equation is often explicitly solvable via the method of characteristics, yielding sufficiently smooth classical solution to establish a verification theorem and confirm the optimality of feedback controls. In such settings, we find that the investor's information acquisition strategy is deterministic and could be decoupled from her trading strategy, indicating a weaker separation property. In some degenerate cases where classical solutions may fail, semi-explicit optimal controls remain attainable by regularizing the information cost.
Liang et al. (2025) studied this question.