G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion

TL;DR

This paper explores G-Brownian motion using rough path theory, establishing pathwise solutions for SDEs and RDEs, and extending these concepts to manifolds with applications to approximation methods.

Contribution

It introduces the enhancement of G-Brownian motion paths to rough paths and develops a framework for SDEs on manifolds driven by G-Brownian motion.

Findings

G-Brownian motion paths can be enhanced to geometric rough paths of roughness 2 < p < 3.

Established the relation between SDEs and RDEs driven by G-Brownian motion.

Developed Euler-Maruyama approximation for G-Brownian motion driven SDEs.

Abstract

The present paper is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the view of rough path theory. As the starting point, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the fundamental relation between SDEs and RDEs driven by G-Brownian motion. As an application, we introduce the notion of SDEs on a differentiable manifold driven by GBrownian motion and construct solutions from the RDE point of view by using pathwise localization technique. This is the starting point of introducing G-Brownian motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin. The last part of this paper is devoted to such construction for a wide and interesting class of G-functions whose invariant group is the orthogonal group. We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian motion of independent interest.