Optimal control of an Ornstein-Uhlenbeck process

In this paper, we consider an Ornstein-Uhlenbeck process in both a finite and a semi-infinite interval. Depending on the form of the cost function, our aim is either to leave the interval as soon as possible or to maximize the time spent in the interval, taking into account the control costs in both cases. The model may represent the current in a simple electrical circuit. Since the exact solutions are in terms of special functions, approximate solutions are given. The deterministic cases are also solved.