Optimally Stopping the Sample Mean of a Wiener Process with an Unknown Drift

It is well-known that optimally stopping the sample mean W(t)/t 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)+Ot with () normally distributed N(JL,(]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>O and b, is equivalent to the time-truncated version of the original problem or the one-arm bandit problem depending on whether (]2a-1. Furthermore, the optimal stopping region changes drastically as the prior parameters (JL,(]2) are slightly perturbed in a neighborhood of (b/a,1/a).