Abstract The jump behavior of an infinitely active Ito semimartingale can be conveniently characterized by a jump activity index of Blumenthal–Getoor type, typically assumed to be constant in time. We study Markovian semimartingales with a non-constant, state-dependent jump activity index and a non-vanishing continuous diffusion component. A nonparametric estimator for the functional jump activity index is proposed and shown to be asymptotically normal under combined high-frequency and long-time-span asymptotics. Furthermore, we propose a nonparametric drift estimator which is robust to symmetric jumps of infinite variance and infinite variation, and which attains the same asymptotic variance as for a continuous diffusion process. Simulations demonstrate the finite sample behavior of our proposed estimators. The mathematical results are based on a novel uniform bound on the Markov generator of the jump diffusion.

中文翻译：

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估计马尔可夫半鞅的状态依赖跳跃活动和漂移

摘要 无限活跃的 Ito 半鞅的跳跃行为可以方便地通过 Blumenthal-Getoor 类型的跳跃活动指数来表征，通常假设在时间上是常数。我们研究具有非常量、状态相关跳跃活动指数和非消失连续扩散分量的马尔可夫半鞅。提出了功能性跳跃活动指数的非参数估计量，并证明在组合的高频和长时间跨度渐近线下渐近正态。此外，我们提出了一种非参数漂移估计器，它对无限方差和无限变化的对称跳跃具有鲁棒性，并且获得与连续扩散过程相同的渐近方差。模拟证明了我们提出的估计器的有限样本行为。