Optimal sequential change-detection for fractional diffusion-type processes

TL;DR

This paper proves the optimality of the CUSUM change detection test for fractional diffusion processes, including fractional Brownian motion with polynomial drift, under Lorden's criteria.

Contribution

It establishes the optimality of the CUSUM test for a broad class of processes and specifically applies this to fractional diffusion-type processes, including fractional Brownian motion.

Findings

CUSUM test is optimal for fractional diffusion processes.

Optimality holds under a modified Lorden's criterion.

CUSUM also optimizes Lorden's original criterion for fractional Brownian motion with polynomial drift.

Abstract

We consider the problem of detecting an abrupt change in the distribution of a sequentially observed stochastic process. We establish the optimality of the CUSUM test with respect to a modified version of Lorden's criterion for arbitrary processes with continuous paths and apply this general result to the special case of fractional diffusion-type processes. As a by-product, we show that the CUSUM test optimizes Lorden's original criterion when a fractional Brownian motion with Hurst index H adopts a polynomial drift term with exponent H + 1/2 after the change.