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. We demonstrate sound finite-sample properties in a simulation study. 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.

翻译：我们基于带有加性跳跃成分的分数过程的高频观测，开发并研究了一种跳跃检验方法。该分数过程的赫斯特指数未知。内填渐近框架下的渐近理论基于弱相依平稳时间序列的极值理论，并延伸了文献中半鞅情形的相关技术。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。我们证明了在存在跳跃的备择假设下检验的一致性，同时建立了局部备择假设下的收敛速率以及跳跃时刻的一致估计。通过模拟研究，我们验证了该方法良好的有限样本性质。在此过程中，我们表明粗糙分数过程赫斯特指数的推断对跳跃具有鲁棒性。这一发现为日益增长的有关粗糙波动率的文献提供了重要见解。