Predicting the Last Zero of Brownian Motion with Drift

Abstract

Given a standard Brownian motion $B^μ=(B_t^μ)_{0\le t\le T}$ with drift $μ\in IR$ and letting $g$ denote the last zero of $B^μ$ before $T$, we consider the optimal prediction problem V_*=\inf_{0\le τ\le T}\mathsf {E}\:|\:g-τ| 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: τ_*=\inf {t\in [0,T] | B_t^μ \le b_-(t) or B_t^μ \ge b_+(t)} where the function $t\mapsto b_-(t)$ is continuous and increasing on $[0,T]$ with $b_-(T)=0$, the function $t\mapsto b_+(t)$ 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 $b_-=-b_+$ and there is a closed form expression for $b_{\pm}$ as shown in [10] using the method of time change from [4]. The latter method cannot be extended to the case when $μ\ne 0$ and the present paper settles the remaining cases using a different approach.