Single-Person Controlled Diffusions with Discounted Costs

This paper describes a class of single-person controlled one-dimensional diffusion processes where the control is a vector-valued function on the state space, which is a compact interval of the real line. These processes generate costs, and the optimal control problem is to choose an admissible control that minimizes expected discounted costs. The major results are necessary and sufficient conditions for a control to be optimal as well as characterizations of the expected costs corresponding to an optimal control. This paper also established sufficient conditions for the existence of piecewise-continuous optimal controls.