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

Abstract

where the infimum is taken over all stopping times τ of B . 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: τ∗ = inf { 0 ≤ t ≤ 1 | S t −B t ≥ b(t) } where b : [0, 1] → IR is a continuous decreasing function with b(1) = 0 that is characterised as the unique solution to a nonlinear Volterra integral equation. This also yields an explicit formula for V∗ in terms of b . If μ = 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 μ 6= 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.