Optimal Control of the Diffusion Coefficient of a Simple Diffusion Process

We consider the optimal control of a one-dimensional diffusion process over a finite time interval. The process may be controlled by varying the diffusion coefficient. The objective is to maximize the expected value of some function of the state, R, at final time. In this paper we investigate the properties of a particular bang-bang control σ₀, and find necessary and sufficient conditions on R for σ₀ to be optimal.