Predicting the Last Zero of Brownian Motion with Drift

TL;DR

This paper addresses the problem of optimally predicting the last zero of a Brownian motion with drift before a fixed time, by formulating it as a free-boundary problem and deriving explicit solutions for the optimal stopping rule.

Contribution

It introduces a novel approach to solve the last zero prediction problem for Brownian motion with non-zero drift, extending previous methods applicable only when the drift is zero.

Findings

Optimal stopping time characterized by boundary functions $b_-(t)$ and $b_+(t)$.

Explicit formula for the value function $V_*$ in terms of these boundaries.

Solution involves coupled nonlinear Volterra integral equations.

Abstract

Given a standard Brownian motion $B^{\mu}=(B_t^{\mu})_{0\le t\le T}$ with drift $\mu \in IR$ and letting $g$ denote the last zero of $B^{\mu}$ before $T$, we consider the optimal prediction problem V_*=\inf_{0\le \tau \le T}\mathsf {E}\:|\:g-\tau | 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 {t\in [0,T] | B_t^{\mu} \le b_-(t) or B_t^{\mu} \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 $\mu=0$ 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 $\mu \ne 0$ and the present paper settles the remaining cases using a different approach.