Cusum tests for changes in the Hurst exponent and volatility of fractional Brownian motion
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In this note, 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 Bégyn (2007) and Nourdin and Nualart (2018, arXive:1808.02378), 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.
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