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

Abstract

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) + θt with θ normally distributed N(μ, σ 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 σ 2 < a −1 or σ 2 > a −1. Furthermore, the optimal stopping region changes drastically as the prior parameters (μ, σ 2) are slightly perturbed in a neighborhood of ( b a , 1 a ).