2013 / March Volume 8 No.1
Doob-Meyer for Rough Paths
| Published Date |
2013 / March
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|---|---|
| Title | Doob-Meyer for Rough Paths |
| Author | |
| Keyword | |
| Download | |
| Pagination | 73-84 |
| 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].
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| AMS Subject Classification |
60H07, 60G15
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| Received |
2012-09-19
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| Accepted |
2012-09-20
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