Optimally stopping the sample mean of a wiener process with an unknown drift

It is well known that optimally stopping the sample mean of a standard Wiener process is associated with a square root boundary. It is shown that when W(t) is replaced by X(t) = W(t) + [theta]t with [theta] normally distributed N([mu], [sigma]2) and independently of the Wiener process, the optimal stopping problem is equivalent to the time-truncated version of the original problem. It is also shown that the problem of optimally stopping (b + X(t))/(a + t), with constants a > 0 and b, is equivalent to the time-truncated version of the original problem or the one-arm bandit problem depending on whether [sigma]2 < a-1 or [sigma]2 > a-1. Furthermore, the optimal stopping region changes drastically as the prior parameters ([mu], [sigma]2) are slightly perturbed in a neighborhood of (, ).