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.

翻译：本文针对具有加性跳跃成分的分数过程，基于高频观测数据开发并研究了一种跳跃检验方法。其中分数过程的Hurst指数未知。在渐进填充理论框架下，我们基于弱相依平稳时间序列的极值理论建立了渐近理论，并拓展了文献中半鞅情形下的技术方法。研究表明，在无跳跃的原假设下，检验统计量弱收敛于Gumbel分布。我们证明了存在跳跃的备择假设下检验的一致性，并建立了局部备择假设下的收敛速度以及跳跃时间的一致估计。在此过程中，我们证明了粗糙分数过程Hurst指数的推断对跳跃具有鲁棒性。这一发现为日益增长的粗糙波动率文献提供了重要见解。通过模拟研究验证了有限样本下的良好性质，并在波动率时间序列的实证案例中展示了方法的适用性。