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

  • Xi Geng
  • Zhongmin Qian
  • Danyu Yang
Part of the Lecture Notes in Mathematics book series (LNM, volume 2123)


The present article is devoted to the study of sample paths of G-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 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 G-Brownian motion and construct solutions from the RDE point of view by using pathwise localization technique. This is the starting point of developing G-Brownian motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin. The last part of this article is devoted to such construction for a wide and interesting class of G-functions whose invariant group is the orthogonal group. In particular, we establish the generating nonlinear heat equation for such G-Brownian motion on a Riemannian manifold. We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian motion of independent interest.


Euler-Maruyama approximation G-Brownian motion G-expectation Geometric rough paths Nonlinear diffusion processes Nonlinear heat flow Rough differential equations 

AMS classification (2010):

60H10 34A12 58J65 



The authors wish to thank Professor Shige Peng for so many valuable suggestions on the present article. The authors are supported by the Oxford-Man Institute at the University of Oxford.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.Oxford-Man InstituteUniversity of OxfordOxfordUK
  3. 3.Exeter CollegeUniversity of OxfordOxfordUK

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