论文标题
在功能数据的背景下实现SPDE的可行中心限制定理
A feasible central limit theorem for realised covariation of SPDEs in the context of functional data
论文作者
论文摘要
本文在无限维度中建立了一种渐近理论，用于估计波动率。我们考虑了半连续性部分偏微分方程的温和溶液，并得出了半群调整后的已实现协变量（SARCV）的稳定中心限制，这是综合波动率的一致估计量，并且是实现的四边形协调对Hilbert空间的概括。此外，我们介绍了半群调整后的多重变体（SAMPV），并确定其较弱的法律。使用SAMPV，我们构建了出现在SARCV中心极限定理中的混合限制过程的渐近协方差的一致估计量，从而导致可行的渐近理论。最后，即使仅在离散的时空网格上可用观察值，我们即使观察结果才能应用结果。
This article establishes an asymptotic theory for volatility estimation in an infinite-dimensional setting. We consider mild solutions of semilinear stochastic partial differential equations and derive a stable central limit theorem for the semigroup adjusted realised covariation (SARCV), which is a consistent estimator of the integrated volatility and a generalisation of the realised quadratic covariation to Hilbert spaces. Moreover, we introduce semigroup adjusted multipower variations (SAMPV) and establish their weak law of large numbers; using SAMPV, we construct a consistent estimator of the asymptotic covariance of the mixed-Gaussian limiting process appearing in the central limit theorem for the SARCV, resulting in a feasible asymptotic theory. Finally, we outline how our results can be applied even if observations are only available on a discrete space-time grid.
