Predicting the time of the ultimate maximum for Brownian motion with drift

Abstract

Given a standard Brownian motion Bμ = (Btμ)0≤t≤1 with drift μ ∈ IR, letting S tμ = max0≤s≤t Bsμ for t ∈ [0, 1], and denoting by θ the time at which S1μ is attained, we consider the optimal prediction problem V* = inf0≤τ≤1 E|θ - τ | where the infimum is taken over all stopping times τ of Bμ. 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| Stμ - Btμ ≥ 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 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 ≠ 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 Lévy processes. © 2008 Springer-Verlag Berlin Heidelberg.