G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion
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.
KeywordsEuler-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.
- 5.G. de Rham, Riemannian Manifolds (Springer, New York, 1984)Google Scholar
- 8.K.D. Elworthy, Stochastic Differential Equations on Manifolds (Springer, New York, 1998)Google Scholar
- 13.N. Ikeda, S.Watanabe, Stochastic Differential Equations and Diffusion Processes (North-Holland, Amsterdam, 1989)Google Scholar
- 16.S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, I and II (Interscience, New York, 1963/1969)Google Scholar
- 22.É. Pardoux, S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic Partial Differential Equations and Their Applications (Springer, New York, 1992), pp. 200–217Google Scholar
- 24.S. Peng, Backward SDE and related g-expectation, in Pitman Research Notes in Mathematics series, vol. 364, ed. by N. El Karoui, L. Mazliak (1997), pp. 141–159Google Scholar
- 25.S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type. Stochast. Anal. Appl. 2, 541–567 (2007)Google Scholar
- 27.S. Peng, Nonlinear expectations and stochastic calculus under uncertainty. Preprint, arXiv: 1002.4546 (2010)Google Scholar