From random walks to rough paths

TL;DR

This paper extends Donsker's invariance principle to rough path topology, enabling new weak limit theorems for stochastic integrals and differential equations driven by random walks.

Contribution

It introduces a novel application of rough path theory to Donsker's invariance principle, broadening its scope to stochastic calculus.

Findings

Donsker's invariance principle holds in rough path topology.

Established weak limit theorems for stochastic integrals.

Extended invariance principles to differential equations driven by random walks.

Abstract

Donsker's invariance principle is shown to hold for random walks in rough path topology. As application, we obtain Donsker-type weak limit theorems for stochastic integrals and differential equations.