Predicting the Time of the Ultimate Maximum for Brownian Motion with Drift

Given a standard Brownian motion B-mu = (B-t(mu))0 <= t <= 1 with drift p E R, letting S-t(mu) = max 0 <= s <= t B-s(mu) for t epsilon [0, 1), and denoting by B the time at which S-1(mu) is attained, we consider the optimal prediction problem V-* = (0 <= r <= 1)inf E vertical bar theta - tau vertical bar where the infimum is taken over all stopping times tau of B-mu. Reducing the optimal prediction problem to a parabolic free-boundary problem and making use of local time-space calculus techniques, we show that the following stopping time is optimal: tau(*) = inf{0 <= t <= 1 vertical bar S-t(mu) - B-t(mu) >= b(t)} where b : [0, 1] -> R is a continuous decreasing function with b(1) = 0 that is characterized as the unique solution to a nonlinear Volterra integral equation. This also yields an explicit formula for V-* in terms of b. If mu = 0 then there is a closed form expression for b. This problem was solved in [14] and [4] using the method of time change. The latter method cannot be extended to the case when mu not equal 0 and the present paper settles the remaining cases using a different approach. It is also shown that the shape of the optimal stopping set remains preserved for all Levy processes.