Control of a Diffusion by Switching between Two Drift-Diffusion Coefficient Pairs

We consider the control of a particular one-dimensional diffusion process over a finite time interval $[0,T]$. Two drift-diffusion pairs $(\mu _1 ,\sigma _1 )$ and $(\mu _2 ,\sigma _2 )$ are given and the process is controlled by switching between these pairs. The objective is to maximize the probability that the process lies in the half line $[0,\infty )$ at final time T.The case where $\mu _1 = \mu _2 = 0$ is considered first. Let the control $\sigma _0 $ be given by the rule: choose the smaller diffusion coefficient if and only if the current state is nonnegative. A result is proved which, loosely speaking, says that $\sigma _0 $ is optimal in this special case, and remains optimal even if we know the final value of the driving Brownian motion in advance.The general problem (with drift) is then solved by an application of this result and the Girsanov transformation.