# Cusum tests for changes in the Hurst exponent and volatility of fractional Brownian motion

## TL;DR

This paper develops cusum change-point tests for detecting shifts in the Hurst exponent and volatility of fractional Brownian motion, ensuring consistency and asymptotic distribution under no change, with practical finite-sample validation.

## Contribution

It introduces a novel cusum testing methodology for fractional Brownian motion parameters, leveraging functional Breuer-Major theorems for theoretical validation and practical feasibility.

## Key findings

- Tests are consistent and asymptotically distribution-free under the no-change hypothesis.
- The methodology accurately estimates change points in finite samples.
- Simulation and data analysis confirm the tests' effectiveness.

## Abstract

In this letter, we construct cusum change-point tests for the Hurst exponent and the volatility of a discretely observed fractional Brownian motion. As a statistical application of the functional Breuer-Major theorems by B\'egyn (2007) and Nourdin and Nualart (2019), we show under infill asymptotics consistency of the tests and weak convergence to the Kolmogorov-Smirnov law under the no-change hypothesis. The test is feasible and pivotal in the sense that it is based on a statistic and critical values which do not require knowledge of any parameter values. Consistent estimation of the break date under the alternative hypothesis is established. We demonstrate the finite-sample properties in simulations and a data example.