Predicting the last zero of Brownian motion with drift

Abstract

Given a standard Brownian motion with drift μ ∈ IR and letting g denote the last zero of before T, we consider the optimal prediction problem where the infimum is taken over all stopping times τ of B μ. 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: where the function t↦b − (t) is continuous and increasing on [0, T] with b − (T) = 0, the function t↦b +(t) is continuous and decreasing on [0, T] with b +(T) = 0, and the pair b − and b + can be characterised as the unique solution to a coupled system of nonlinear Volterra integral equations. This also yields an explicit formula for V * in terms of b − and b +. If μ = 0 then and there is a closed form expression for b ± as shown in Shiryaev (Theory Probab. Appl. in press) using the method of time change from Graversen et al. (2001, Theory Probab. Appl. 45, 125–136). The latter method cannot be extended to the case when μ ≠ 0 and the present paper settles the remaining cases using a different approach.