A quasi-sure non-degeneracy property for the Brownian rough path

Author: 

Boedihardjo, H
Geng, X
Liu, X
Qian, Z

Publication Date: 

7 May 2018

Journal: 

Potential Analysis

Last Updated: 

2020-01-22T00:16:25.237+00:00

Issue: 

1

Volume: 

51

DOI: 

10.1007/s11118-018-9699-1

page: 

1–21-

abstract: 

In the present paper, we are going to show that outside a slim set in the sense of Malliavin (or quasi-surely), the signature path (which consists of iterated path integrals in every degree) of Brownian motion is non-self-intersecting. This property relates closely to a non-degeneracy property for the Brownian rough path arising naturally from the uniqueness of signature problem in rough path theory. As an important consequence we conclude that quasi-surely, the Brownian rough path does not have any tree-like pieces and every sample path of Brownian motion is uniquely determined by its signature up to reparametrization.

Symplectic id: 

835665

Submitted to ORA: 

Not Submitted

Publication Type: 

Journal Article