Regularization Effects of Time Integration on Gaussian Process Functionals

TL;DR

This paper explores how time integration influences the regularization of Gaussian process functionals, using Malliavin calculus to analyze covariance functions and their smoothness properties.

Contribution

It introduces sufficient conditions for regularization of Gaussian functionals via time integration and examines their implications for level-crossing smoothness.

Findings

Identifies conditions under which covariance functions are regularized by time integration

Demonstrates regularization effects on important classes of Gaussian process covariances

Provides insights into the smoothness of level-crossing functionals

Abstract

In this paper, we investigate the regularization effects, in the sense of Malliavin calculus, on functionals of Gaussian processes induced by time integration, focusing on their covariance functions. We study several examples of important covariance functions classes to verify whether they satisfy the sufficient conditions proposed for regularization. Additionally, we derive a weak implication for the smoothness of level-crossing functionals.