The Geometry of Rough Path Space

TL;DR

This paper introduces a vector space structure on a subset of p-rough paths, enabling additive perturbations and extending the algebraic framework of rough path theory.

Contribution

It defines a vector space $H^p(V)$ within rough path space, establishes algebraic properties, and shows enlarging to almost rough paths does not increase displacement sets.

Findings

$H^p(V)$ forms a vector space under $oxplus$ and $ullet$

Additive perturbations of rough paths are well-defined and associative

Enlarging to almost rough paths does not expand displacement sets

Abstract

We describe $H^p(V)$, a subset of $p$-rough path space $\Omega_p(V)$ which is a vector space under an addition operation $\boxplus$ and a scalar multiplication $\odot$. We show that the domain of $\boxplus$ can be extended to $\Omega_p(V)\times H^p(V)$, allowing any $p$-rough path $X$ to be additively perturbed by an $H\in H^p(V)$. We prove associativity $(X\boxplus H)\boxplus \tilde H = X\boxplus (H\boxplus \tilde H)$ and trivial kernel $X\boxplus H = X \Leftrightarrow H = 1$, where $1$ is the additive zero in $(H^p(V),\boxplus,\odot)$. Finally, we show that enlarging $H^p(V)$ to almost rough paths $H^{am,p}(V)$ does not enlarge the set of displacements of a given $X$, i.e. $\{X\boxplus H: H\in H^p(V)\}=\{X\boxplus H: H\in H^{am,p}(V)\}$.