On optimal control of a brownian motion

This paper is concerned with the optimal control of a Brownian motion on R⁺. The process is controlled by switching control action to affect the variance and the drift. There is a cost which depends on the state of the process and the action used to operate the system. There is another cost incurred instantaneously to switch the control action. A quasi-variational inequality is solved as the dynamic programming equation for a long run average cost criterion.