A note on the notion of geometric rough paths

By Peter K. Friz, Nicolas Victoir. 2004

74 citations recorded by the source corpus.

We use simple sub-Riemannian techniques to prove that every weak geometric p-rough path (a geometric p-rough path in the sense of [20]) is the limit in sup-norm of a sequence of canonically lifted smooth paths, uniformly bounded in p-variation, thus clarifying the two different definitions of a geometric p-rough path. Our proofs are sufficiently general to include the case of Hölder- and modulus-type regularity. This allows us to extend a few classical results on Hölder-spaces and p -variation spaces to the non-commutative setting necessary for the theory of rough paths. As an application, we give a precise description of the support of Enhanced Fractional Brownian Motion, and prove a conjecture by Ledoux et al.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/695ff926a95b64e3afd5fbd572123897761cca55

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In Applied Mathematics

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Citations in the corpus

Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, background): “2 appeared in [59]; here it is derived as a special case of a general approximation result on geodesic spaces.”
Non-degeneracy of Wiener functionals arising from rough differential equations (2007, background): “Using the Gauss tail of the homogenous p -variation norm of Gaussian rough paths (see [7, 8]) we see that L q -estimates for all q ∞ hold true when p < 2, and this underlies to density results of [26, 1].”
Flows driven by rough paths (2012, background): “A Hölder weak geometric p-rough path with the above property is called a Hölder geometric p-rough path; not all Hölder weak geometric p-rough path are Hölder geometric p-rough path [8].”
Densities for rough differential equations under Hormander's condition (2007, background): “[28] (who obtained it for the Brownian rough path) and carried out independently in [11, 15] for H > 1/3.”
An extension theorem to rough paths (2007, background): “There are examples of weak geometric p-rough paths that are not geometric p-rough paths [6].”
A theory of regularity structures (2013, unknown): “(See also the note [43].”
Differential equations driven by Gaussian signals (2007, background): “We recall the following simple interpolation result [13].”
Rough path limits of the Wong–Zakai type with a modified drift term (2009, background): “uous. It is a wellknown [Lyo98] that x can be lifted uniquely to a α-Holder path SN (x) with values in GN Rd . Then, SN (x)0,·⊗p ∈C min(α,β)-Ho¨l [0,T],GN Rd . From general facts of such spaces (e.g. [FV06b], relying only on the fact that GN Rd is a geodesic space) we can ﬁnd a sequence of Lipschitz”