Bayesian sequential testing of the drift of a Brownian motion

TL;DR

This paper addresses Bayesian sequential testing of a Brownian motion's drift sign, reformulating it as an optimal stopping problem and characterizing the optimal boundaries for finite and infinite horizons.

Contribution

It introduces a novel reformulation of the Bayesian drift testing as an optimal stopping problem and provides a complete characterization of the optimal boundaries.

Findings

Boundaries are monotonic and continuous in finite-horizon cases.

Explicit integral equations characterize the optimal stopping boundaries.

A convergent scheme approximates boundaries in the infinite-horizon case.

Abstract

We study a classical Bayesian statistics problem of sequentially testing the sign of the drift of an arithmetic Brownian motion with the $0$-$1$ loss function and a constant cost of observation per unit of time for general prior distributions. The statistical problem is reformulated as an optimal stopping problem with the current conditional probability that the drift is non-negative as the underlying process. The volatility of this conditional probability process is shown to be non-increasing in time, which enables us to prove monotonicity and continuity of the optimal stopping boundaries as well as to characterize them completely in the finite-horizon case as the unique continuous solution to a pair of integral equations. In the infinite-horizon case, the boundaries are shown to solve another pair of integral equations and a convergent approximation scheme for the boundaries is provided. Also, we describe the dependence between the prior distribution and the long-term asymptotic behaviour of the boundaries.