Related papers: H\"older-continuous rough paths by Fourier normal …
Fourier normal ordering \cite{Unt09bis} is a new algorithm to construct explicit rough paths over arbitrary H\"older-continuous multidimensional paths. We apply in this article the Fourier normal ordering ordering algorithm to the…
We construct in this article a rough path over fractional Brownian motion with arbitrary Hurst index by (i) using the Fourier normal ordering algorithm introduced in \cite{Unt-Holder} to reduce the problem to that of regularizing tree…
We construct an explicit Hopf algebra isomorphism from the algebra of heap-ordered trees to that of quasi-symmetric functions, generated by formal permutations, which is a lift of the natural projection of the Connes-Kreimer algebra of…
The main tool for stochastic calculus with respect to a multidimensional process $B$ with small H\"older regularity index is rough path theory. Once $B$ has been lifted to a rough path, a stochastic calculus -- as well as solutions to…
In this paper, we show how one can view certain models in regularity structures as some form of geometric rough paths. This is performed by identifying the deformed Butcher-Connes-Kreimer Hopf algebra with a quotient of the shuffle Hopf…
The stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended Chen's multiplicative property involves the D\"urr-Connes-Kreimer…
We build a connection between rough path theory and noncommutative algebra, and interpret the integration of geometric rough paths as an example of a non-abelian Young integration. We identify a class of slowly-varying one-forms, and prove…
Lyon's rough paths give an algebraic and analytic framework for Stieltjes integrals in a regime of low regularity where the usual Riemann-Stieltjes integral does not converge. Before we may rigorously define rough paths, we start with the…
In this article we extend the framework of rough paths to processes of variable H\"older exponent or variable order paths. We show how a class of multiple discrete delay differential equations driven by signals of variable order are…
Using some basic notions from the theory of Hopf algebras and quasi-shuffle algebras, we introduce rigorously a new family of rough paths: the quasi-geometric rough paths. We discuss their main properties. In particular, we will relate them…
We consider the rough differential equation with drift driven by a Gaussian geometric rough path. Under natural conditions on the rough path, namely non-determinism, and uniform ellipticity conditions on the diffusion coefficient, we prove…
We prove an extension to the classical continuity theorem in rough paths. We show that two $p$-rough paths are close in all levels of iterated integrals provided the first $\lfl p \rfl$ terms are close in a uniform sense. Applications…
Optimal sample path properties of stochastic processes often involve generalized H\"{o}lder- or variation norms. Following a classical result of Taylor, the exact variation of Brownian motion is measured in terms of $\psi (x) \equiv $…
Starting from the construction of a geometric rough path associated with a fractional Brownian motion with Hurst parameter $H\in]{1/4}, {1/2}[$ given by Coutin and Qian (2002), we prove a large deviation principle in the space of geometric…
Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study of algebraic structures with replicating operations. To understand their algebraic and combinatorial nature, we first use rooted…
Within the context of rough path analysis via fractional calculus, we show how variability can be used to prove the existence of integrals with respect to H\"older continuous multiplicative functionals in the case of Lipschitz coefficients…
The aim of the paper is to show the probabilistically strong well-posedness of rough differential equations with distributional drifts driven by the Gaussian rough path lift of fractional Brownian motion with Hurst parameter…
We construct a canonical geometric rough path over $d$-dimensional tempered fractional Brownian motion (tfBm) for any Hurst parameter $H > 1/4$ and tempering parameter $\lambda > 0$. The main challenge stems from the non-homogeneous nature…
Let $B=(B_1(t),..,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha\le 1/4$, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of $B$ is a…
We develop a Fourier approach to rough path integration, based on the series decomposition of continuous functions in terms of Schauder functions. Our approach is rather elementary, the main ingredient being a simple commutator estimate,…