Optimal adaptive control with separable drift uncertainty

We consider a problem of stochastic optimal control with separable drift uncertainty in strong formulation on a finite time horizon. The drift of the state Y u is multiplicatively influenced by an unknown random variable λ, while admissible controls u are required to be adapted to the observation filtration. Choosing a control actively influences the state and information acquisition simultaneously and comes with a learning effect. The problem, initially non-Markovian, is embedded into a higher-dimensional Markovian, full information control problem with control-dependent filtration and noise. To that problem, we apply the stochastic Perron method to characterize the value function as the unique viscosity solution of the HJB equation, explicitly construct ε-optimal controls, and show that the values in the strong and weak formulation agree. Numerical illustrations show a significant difference between the adaptive control and the certainty equivalence control, highlighting a substantial learning effect.