Differential Equations Driven by Gaussian Signals I

TL;DR

This paper develops a new framework for analyzing differential equations driven by Gaussian signals using rough path theory, introducing a sharp covariance condition and novel approximation results.

Contribution

It presents a simple, sharp covariance condition for Gaussian processes, constructs Gaussian rough paths, and introduces a new RKHS embedding for analyzing Gaussian-driven differential equations.

Findings

Established a new covariance condition for Gaussian processes

Constructed Gaussian rough paths with approximation results

Developed a new RKHS embedding for analysis

Abstract

We consider multi-dimensional Gaussian processes and give a new condition on the covariance, simple and sharp, for the existence of stochastic area(s). Gaussian rough paths are constructed with a variety of weak and strong approximation results. Together with a new RKHS embedding, we obtain a powerful - yet conceptually simple - framework in which to analysize differential equations driven by Gaussian signals in the rough paths sense.