Optimal drift on $[0,1]$

Abstract:

Consider one-dimensional diffusions on the interval $[0,1]$ of the form $d{X_t} = d{B_t} + b({X_t})dt$, with $0$ a reflecting boundary, $b(x) \geqslant 0$, and $\int _0^1 {b(x)dx = 1}$. In this paper, we show that there is a unique drift which minimizes the expected time for ${X_t}$ to hit $1$, starting from ${X_0} = 0$. In the deterministic case $d{X_t} = b({X_t})dt$, the optimal drift is the function which is identically equal to $1$. By contrast, if $d{X_t} = d{B_t} + b({X_t})dt$, then the optimal drift is the step function which is $2$ on the interval $[1/4,3/4]$ and is $0$ otherwise. We also solve this problem for arbitrary starting point ${X_0} = {x_0}$ and find that the unique optimal drift depends on the starting point, ${x_0}$, in a curious manner.