Using a geometric Brownian motion to control a Brownian motion and vice versa

Let x(t) be a one-dimensional Brownian motion. The homing problem for a controlled x(t) process is solved by using a mathematical expectation for an uncontrolled geometric Brownian motion. Furthermore, it turns out that the optimally controlled process is a Bessel process. Similarly, a geometric Brownian motion is optimally controlled by using a mathematical expectation for an uncontrolled Brownian motion process.