Statistical analysis of discretely sampled semilinear SPDEs: a power variation approach
Statistical analysis of discretely sampled semilinear SPDEs: a power variation approach
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DOI：
10.1007/s40072-022-00285-3
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发表时间：
2021-03
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影响因子：
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通讯作者：
Igor Cialenco;Hyun-Jung Kim;Gregor Pasemann
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文献类型：
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作者：
Igor Cialenco;Hyun-Jung Kim;Gregor Pasemann
受离散采样随机微分方程统计分析问题的启发，首先推导了高阶有限差分的中心极限定理，该定理适用于具有任意非正则路径的随机过程.这些结果证明了使用的概念，功率变化，这里介绍，沿着与Hölder-Zygmund规范。因此，我们证明了一个新的分数布朗运动的迭代积分的幂变分的中心极限定理。这些抽象的结果，除了是独立的利益，在第二部分的文件中被应用到估计的漂移和波动系数的半线性随机偏微分方程的一维，驱动的加性高斯噪声白色在时间上和可能的颜色在空间中。特别是，我们解决了Cialenco等人（Stat.推理Stoch。过程23：83-103，2020）关于通过有限差分的导数的朴素近似导出的估计量中存在非平凡偏差。我们给出了一个显式的偏差公式，并推导出相应的估计的收敛速度。数值例子说明了理论结果。
Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems for higher order finite differences applied to stochastic processes with arbitrary finitely regular paths. These results are proved by using the notion of-power variations, introduced herein, along with the Hölder-Zygmund norms. Consequently, we prove a new central limit theorem for-power variations of the iterated integrals of a fractional Brownian motion. These abstract results, besides being of independent interest, in the second part of the paper are applied to estimation of the drift and volatility coefficients of semilinear stochastic partial differential equations in dimension one, driven by an additive Gaussian noise white in time and possibly colored in space. In particular, we solve the earlier conjecture from Cialenco et al. (Stat. Inference Stoch. Process. 23:83-103, 2020) about existence of a nontrivial bias in the estimators derived by naive approximations of derivatives by finite differences. We give an explicit formula for the bias and derive the convergence rates of the corresponding estimators. Theoretical results are illustrated by numerical examples.