In this paper we propose a nonparametric test to determine whether an underlying jump diffusion process indeed contains jump component, or equivalently, is indeed a diffusion. Our test is based upon a robust threshold estimation of diffusive volatility and the kernel estimation of the conditional moment function of the squared instantaneous increments of the underlying process. We show that our test statistic has asymptotic standard normal distribution under the null hypothesis of no jumps, is consistent against fixed alternatives, and may detect local alternatives that shrink to diffusions at certain convergence rates, when sampling interval shrinks to zero and time span is either fixed or expands. We only assume that the jump diffusion process is recurrent, thus allowing for both stationary and nonstationary cases. In addition, we provide a regression bootstrap test and establish its validity. A Monte Carlo simulation is conducted to examine the finite sample performances of our test, and an empirical illustration is also provided.

在本文中，我们提出了一种非参数检验来确定潜在的跳跃扩散过程是否确实包含跳跃分量，或者等效地，确实是扩散。我们的测试基于扩散波动率的稳健阈值估计和基础过程平方瞬时增量的条件矩函数的核估计。我们表明，我们的检验统计量在无跳跃的原假设下具有渐近标准正态分布，与固定替代方案一致，并且当采样间隔缩小到零且时间跨度为固定或扩展。我们只假设跳跃扩散过程是循环的，因此允许平稳和非平稳情况。此外，我们提供回归自举测试并确定其有效性。进行蒙特卡罗模拟以检查我们测试的有限样本性能，并提供经验说明。