Doob--Meyer for rough paths

Abstract

Recently, Hairer--Pillai proposed the notion of $\theta$-roughness of a path which leads to a deterministic Norris lemma. In the Gubinelli framework (Hoelder, level 2) of rough paths, they were then able to prove a Hoermander type result (SDEs driven by fractional Brownian motion, $H>1/3$). We take a step back and propose a natural "roughness" condition relative to a given $p$-rough path in the sense of Lyons; the aim being a Doob-Meyer result for rough integrals in the sense of Lyons. The interest in our (weaker) condition is that it is immediately verified for large classes of Gaussian processes, also in infinite dimensions. We conclude with an application to non-Markovian system under Hoermander's condition.

References in corpus (3)

• Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths
• Integrability and tail estimates for Gaussian rough differential equations
• Smoothness of Density for the Area Process of Fractional Brownian Motion

Cited by in corpus (7)

• Smoothness of the density for solutions to Gaussian rough differential equations
• Pathwise integration with respect to paths of finite quadratic variation
• Averaging along irregular curves and regularisation of ODEs
• C-infinity regularization of ODEs perturbed by noise
• High order paracontrolled calculus
• Integration of time-varying cocyclic one-forms against rough paths
• Existence of densities for stochastic evolution equations driven by fractional Brownian motion