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Doob-Meyer for Rough Paths
by Peter Friz   Atul Shekhar

Vol. 8 No. 1 (2013) P.73~P.84

ABSTRACT

Recently Hairer--Pillai [Regularity of Laws and Ergodicity of Hypoelliptic SDEs Driven by Rough Paths, to appear in Annals of Probability] proposed the notion of $\theta$-roughness of a path which leads to a deterministic Norris lemma. In the Gubinelli framework (Hölder, level $2$) of rough paths, they were then able to prove a Hörmander 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 Hörmander's condition, in the spirit of Cass--Friz [Densities for rough differential equations under Hörmander's condition, Ann. of Math. (2), 171(3):2115--2141, 2010].

KEYWORDS
Rough paths, fine properties of stochastic processes, Doob-Meyer decomposition, H\

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 60H07, 60G15

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