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

Boedihardjo, H., Geng, X., Liu, X. and Qian, Z. (2019) A quasi-sure non-degeneracy property for the Brownian rough path. Potential Analysis, 51 (1). pp. 1-21. ISSN 0926-2601

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To link to this item DOI: 10.1007/s11118-018-9699-1


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-selfintersecting. 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.

Item Type:Article
Divisions:Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:76640


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