Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems for higher order finite differences applied to stochastic process with arbitrary finitely regular paths. These results are proved by using the notion of $\Delta$-power variations, introduced herein, along with the H\"older-Zygmund norms. Consequently, we prove a new central limit theorem for $\Delta$-power variations of the iterated integrals of a fractional Brownian motion (fBm). 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, Kim, Lototsky (2019) 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.
翻译:由不同抽样的SPDE的统计分析产生的问题所引发,首先,我们得出了对任意的固定路径进行随机随机随机随机随机随机切换过程应用的更高顺序有限差异的中央限值理论。这些结果通过使用此处引入的美元/德尔塔元-功格蒙德标准以及H\"olderer-Zygmund标准 " 的概念来证明。因此,我们证明对分数布朗运动(fBm)的迭接组合的重迭组合的美元/德尔塔元-功率变化有一个新的中央限值理论。这些抽象结果,除了具有独立的兴趣外,还应用在本文第二部分中用于对一维半线性半线性随机切换部分差异方程式的漂移和波动系数进行估计,其驱动因素是加加加加加高斯时的噪音在时间和空间可能具有色化作用。特别是,我们解决了Cialenco、Kim、Lotsky(2019年)的早先的参数,即根据有限差异对衍生物的日近近近似近似近似的估测中是否存在非微偏差的偏差的偏差,我们解决了早期的近论。我们给出了一个明确的模型,提出了一个明确的公式。我们用直观的一致率的精确度的公式。