A variation embedding theorem and applications

By Peter K. Friz, Nicolas Victoir. 2005

64 citations recorded by the source corpus.

Fractional Sobolev spaces, also known as Besov or Slobodetski spaces, arise in many areas of analysis, stochastic analysis in particular. We prove an embedding into certain q-variation spaces. Applications include a new route to a regularity result by Kusuoka for stochastic differential equations, integration against Besov-paths, a regularity criterion for rough paths and a new regularity result for Cameron–Martin paths associated to fractional Brownian motion.

Subfields: Applied Mathematics, Finance, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/d549e97584c199a4cf44d3192a48713d9743dc0b

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Citations in the corpus

General Rough integration, Levy Rough paths and a Levy--Kintchine type formula (2012, unknown): “(ii) In [12], an embedding result W ↪→Cp-var is shown, more precisely ‖X‖qp-var;[0,T ] ∫ T”
The Signature of a Rough Path: Uniqueness (2014, unknown): “(2) (see [24], Corollary 1) If 0 < H 6 1/2, then for any q > (H + 1/2) , we have H ⊂ C 0 ( [0, 1];R ) .”
Differential equations driven by rough paths with jumps (2017, unknown): “which implies, by a Besov variation embedding theorem ([13], see also [12, Cor.”
Differential equations driven by Gaussian signals (2007, background): “proximations for fractional Brownian Motion is studied in [ 19 ].”
Smoothness of the density for solutions to Gaussian rough differential equations (2012, background): “Writing HH for the Cameron–Martin space of fBm for H in (1/4,1/2), the variation embedding in [14] gives the stronger result that H →֒Cq-var([0, T ],R) for any q > (H + 1/2).”
Integrability Estimates for Gaussian Rough Differential Equations (2011, background): “Writing HH for the Cameron–Martin space of fBm for H in (1/4,1/2), the variation embedding in [9] gives the stronger result that H →֒Cq-var(I,Rd) for any q > (H +1/2).”
Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, background): “In fact, one can do a little better and show that for any  ∈ (12 12 +)  H → 2 0 →   The first embedding is well-known: From [36] and the references therein we know that H is continuously embedded in the potential space  +122, which we need not define here, and from [47, 36] one ha”
Densities for rough differential equations under Hormander's condition (2007, methodology): “Writing HH for the Cameron-Martin space of fBM, the variation embedding in [16] gives H →֒ C for any q > (H + 1/2) .”