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

TL;DR

The paper addresses sequential Bayesian testing for the drift of a fractional Brownian motion with an unknown drift $\theta\sim N(\mu,\sigma^2)$, observing $Z_t=\theta t+B^H_t$ and testing $\theta>0$ vs $\theta\le0$. By transforming the fBm to a diffusion and applying Bayes/optimal stopping theory, the authors derive an exact non-asymptotic optimal test: the stopping rule corresponds to the first exit from a time-varying boundary $A(t)$, which is characterized by a nonlinear integral equation and solvable numerically. The main contributions are (i) a complete reduction to a standard Brownian stopping problem via a Brownian representation and time-space change, (ii) a rigorously defined stopping boundary with existence, continuity, and a unique integral equation, and (iii) a practical numerical scheme for computing the boundary. This yields a principled, exact sequential procedure for deciding the drift sign in a non-Markovian Gaussian setting, with potential applications in fields where fBm models arise.

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.

Figures (1)

• Figure 1: The stopping boundary $A(t)$ for different values of $H$ and $\sigma=1$.