  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.
