Journal article

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

Xi Geng, Zhongmin Qian, Danyu Yang, C DonatiMartin (ed.), A Lejay (ed.), A Rouault (ed.)

SEMINAIRE DE PROBABILITES XLVI | SPRINGER INTERNATIONAL PUBLISHING AG | Published : 2014

Abstract

The present article is devoted to the study of sample paths ofG-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the viewpoint of rough path theory. As the starting point, by using techniques in rough path theory, 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 GBrownian 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-Bro..

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