在希尔伯特空间环境中实现共变的大量法律的软弱无力 (A weak law of large numbers for realised covariation in a Hilbert space setting)

This article generalises the concept of realised covariation to Hilbert-space-valued stochastic processes. More precisely, based on high-frequency functional data, we construct an estimator of the trace-class operator-valued integrated volatility process arising in general mild solutions of Hilbert space-valued stochastic evolution equations in the sense of Da Prato and Zabczyk (2014). We prove a weak law of large numbers for this estimator, where the convergence is uniform on compacts in probability with respect to the Hilbert-Schmidt norm. In addition, we show that the conditions on the volatility process are valid for most common stochastic volatility models in Hilbert spaces.

翻译：本文概括了Hilbert-space-valited schoestic 进程实现共变的概念。更准确地说,根据高频功能数据,我们构建了一个测算器,用于测算在Hilbert Da Prato和Zabczyk (2014年)意义上的Hilbert 空间价值的随机进化方程式的一般温和解决方案中产生的微粒级操作者估值的综合波动过程。对于这个测算器来说,我们证明我们是一个大量存在的薄弱法律,因为对于Hilbert-Schmidt 规范而言,在契约上的趋同程度是统一的。此外,我们表明,在Hilbert 空间最常见的随机变异模型中,关于波动过程的条件是有效的。