A controller-stopper-game with hidden controller type

We consider a continuous time stochastic dynamic game between a stopper (the owner of an asset) and a controller (the manager) who is either effective or non-effective. An effective manager can exert high or low effort which corresponds to a high or a low positive drift for the accumulated income of the owner with random noise in terms of Brownian motion. The manager earns a salary until the owner stops the game. A non-effective manager cannot act but receives a salary. We find a threshold (Nash) equilibrium using stochastic filtering methods in a weak formulation.