A Bayesian sequential test for the drift of a fractional Brownian motion

TL;DR

This paper develops a Bayesian sequential testing method for detecting the sign of the drift in a fractional Brownian motion, transforming it into an optimal stopping problem for standard Brownian motion.

Contribution

It introduces a novel approach by reducing the fractional Brownian motion drift test to an optimal stopping problem, with a numerical solution for the boundary conditions.

Findings

The test reduces to an optimal stopping problem for Brownian motion.

Boundaries of the stopping set satisfy a specific integral equation.

Numerical methods are used to solve the integral equation.

Abstract

We consider a fractional Brownian motion with unknown linear drift such that the drift coefficient has a prior normal distribution and construct a sequential test for the hypothesis that the drift is positive versus the alternative that it is negative. We show that the problem of constructing the test reduces to an optimal stopping problem for a standard Brownian motion, obtained by a transformation of the fractional one. The solution is described as the first exit time from some set, whose boundaries are shown to satisfy a certain integral equation, which is solved numerically.