Testing for jumps in processes with integral fractional part and jump-robust inference on the Hurst exponent

We develop and investigate a test for jumps based on high-frequency observations of a fractional process with an additive jump component. The Hurst exponent of the fractional process is unknown. The asymptotic theory under infill asymptotics builds upon extreme value theory for weakly dependent, stationary time series and extends techniques for the semimartingale case from the literature. It is shown that the statistic on which the test is based on weakly converges to a Gumbel distribution under the null hypothesis of no jumps. We prove consistency under the alternative hypothesis when there are jumps. Moreover, we establish convergence rates for local alternatives and consistent estimation of jump times. In the process, we show that inference on the Hurst exponent of a rough fractional process is robust with respect to jumps. This provides an important insight for the growing literature on rough volatility. We demonstrate sound finite-sample properties in a simulation study and showcase the applicability of our methods in an empirical example with a time series of volatilities.