High-frequency analysis of parabolic stochastic PDEs

TL;DR

This paper develops high-frequency statistical methods for estimating the volatility and spatial covariance structure of solutions to parabolic stochastic PDEs, enabling more accurate modeling of complex stochastic systems.

Contribution

It introduces nonparametric estimators based on limit theorems for multipower variations, applicable to high-frequency data of stochastic PDE solutions.

Findings

Consistent estimators for integrated volatility.

Asymptotic confidence bounds for volatility.

Feasible estimators for spatial covariance regularity.

Abstract

We consider the problem of estimating stochastic volatility for a class of second-order parabolic stochastic PDEs. Assuming that the solution is observed at a high temporal frequency, we use limit theorems for multipower variations and related functionals to construct consistent nonparametric estimators and asymptotic confidence bounds for the integrated volatility process. As a byproduct of our analysis, we also obtain feasible estimators for the regularity of the spatial covariance function of the noise.