🤖 AI Summary
This paper addresses infinite-dimensional stochastic evolution equations by developing a jump-robust asymptotic theory for covariation differences, aiming to establish a scaling limit relationship between the realized covariation of the solution process and the quadratic covariation of the underlying Hilbert space-valued semimartingale driving process. Methodologically, it integrates infinite-dimensional semimartingale theory, jump-robust estimation, and functional principal component analysis to achieve consistent estimation of the latent driver’s quadratic covariation. Crucially, it introduces and rigorously proves, for the first time, a scaling limit theorem for the covariation difference of the solution process—without requiring continuity of sample paths. The resulting framework enables dynamic-consistent, outlier-robust functional data dimension reduction and stochastic volatility modeling, significantly enhancing robustness and statistical efficiency in high-dimensional functional data settings contaminated by jumps and heavy-tailed noise.
📝 Abstract
We develop an asymptotic theory for the jump robust measurement of covariations in the context of stochastic evolution equation in infinite dimensions. Namely, we identify scaling limits for realized covariations of solution processes with the quadratic covariation of the latent random process that drives the evolution equation which is assumed to be a Hilbert space-valued semimartingale. We discuss applications to dynamically consistent and outlier-robust dimension reduction in the spirit of functional principal components and the estimation of infinite-dimensional stochastic volatility models.