With the aim of improving the reconstruction of stochastic evolution equations from empirical time-series data, we derive a full representation of the generator of the Kramers–Moyal operator via a power-series expansion of the exponential operator. This expansion is necessary for deriving the different terms in a stochastic differential equation. With the full representation of this operator, we are able to separate finite-time corrections of the power-series expansion of arbitrary order into terms with and without derivatives of the Kramers–Moyal coefficients. We arrive at a closed-form solution expressed through conditional moments, which can be extracted directly from time-series data with a finite sampling intervals. We provide all finite-time correction terms for parametric and non-parametric estimation of the Kramers–Moyal coefficients for discontinuous processes which can be easily implemented—employing Bell polynomials—in time-series analyses of stochastic processes. With exemplary cases of insufficiently sampled diffusion and jump-diffusion processes, we demonstrate the advantages of our arbitrary-order finite-time corrections and their impact in distinguishing diffusion and jump-diffusion processes strictly from time-series data.

中文翻译：

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Kramers-Moyal 算子的任意阶有限时间修正


为了改进从经验时间序列数据重建随机演化方程的目的，我们通过指数算子的幂级数展开导出了 Kramers-Moyal 算子生成元的完整表示。这种展开对于导出随机微分方程中的不同项是必要的。通过该算子的完整表示，我们能够将任意阶幂级数展开的有限时间校正分离为具有和不具有 Kramers-Moyal 系数导数的项。我们得到了通过条件矩表示的封闭式解，它可以直接从具有有限采样间隔的时间序列数据中提取。我们为不连续过程的 Kramers-Moyal 系数的参数和非参数估计提供了所有有限时间修正项，这些修正项可以在随机过程的时间序列分析中轻松实现（采用贝尔多项式）。通过采样不足的扩散和跳跃扩散过程的示例案例，我们证明了任意阶有限时间校正的优势及其在严格区分扩散和跳跃扩散过程与时间序列数据方面的影响。