\(G\)-Brownian motion as rough paths and differential equations driven by \(G\)-Brownian motion. (English) Zbl 1390.60205
Donati-Martin, Catherine (ed.) et al., Séminaire de Probabilités XLVI. Cham: Springer (ISBN 978-3-319-11969-4/pbk; 978-3-319-11970-0/ebook). Lecture Notes in Mathematics 2123. Séminaire de Probabilités, 125-193 (2014).
Summary: 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.
For the entire collection see [Zbl 1305.60008].
For the entire collection see [Zbl 1305.60008].
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60J65 | Brownian motion |
| 60J60 | Diffusion processes |
Keywords:
Euler-Maruyama approximation; \(G\)-Brownian motion; \(G\)-expectation; geometric rough paths; nonlinear diffusion processes; nonlinear heat flow; rough differential equationsReferences:
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