Ergodicity of hypoelliptic SDEs driven by fractional brownian motion

M. Hairer; N. S. Pillai

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 2, page 601-628
  • ISSN: 0246-0203

Abstract

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We demonstrate that stochastic differential equations (SDEs) driven by fractional brownian motion with Hurst parameter H>½ have similar ergodic properties as SDEs driven by standard brownian motion. The focus in this article is on hypoelliptic systems satisfying Hörmander’s condition. We show that such systems enjoy a suitable version of the strong Feller property and we conclude that under a standard controllability condition they admit a unique stationary solution that is physical in the sense that it does not “look into the future.” The main technical result required for the analysis is a bound on the moments of the inverse of the Malliavin covariance matrix, conditional on the past of the driving noise.

How to cite

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Hairer, M., and Pillai, N. S.. "Ergodicity of hypoelliptic SDEs driven by fractional brownian motion." Annales de l'I.H.P. Probabilités et statistiques 47.2 (2011): 601-628. <http://eudml.org/doc/242429>.

@article{Hairer2011,
abstract = {We demonstrate that stochastic differential equations (SDEs) driven by fractional brownian motion with Hurst parameter H&gt;½ have similar ergodic properties as SDEs driven by standard brownian motion. The focus in this article is on hypoelliptic systems satisfying Hörmander’s condition. We show that such systems enjoy a suitable version of the strong Feller property and we conclude that under a standard controllability condition they admit a unique stationary solution that is physical in the sense that it does not “look into the future.” The main technical result required for the analysis is a bound on the moments of the inverse of the Malliavin covariance matrix, conditional on the past of the driving noise.},
author = {Hairer, M., Pillai, N. S.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {ergodicity; fractional brownian motion; Hörmander’s theorem; fractional Brownian motion},
language = {eng},
number = {2},
pages = {601-628},
publisher = {Gauthier-Villars},
title = {Ergodicity of hypoelliptic SDEs driven by fractional brownian motion},
url = {http://eudml.org/doc/242429},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Hairer, M.
AU - Pillai, N. S.
TI - Ergodicity of hypoelliptic SDEs driven by fractional brownian motion
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 2
SP - 601
EP - 628
AB - We demonstrate that stochastic differential equations (SDEs) driven by fractional brownian motion with Hurst parameter H&gt;½ have similar ergodic properties as SDEs driven by standard brownian motion. The focus in this article is on hypoelliptic systems satisfying Hörmander’s condition. We show that such systems enjoy a suitable version of the strong Feller property and we conclude that under a standard controllability condition they admit a unique stationary solution that is physical in the sense that it does not “look into the future.” The main technical result required for the analysis is a bound on the moments of the inverse of the Malliavin covariance matrix, conditional on the past of the driving noise.
LA - eng
KW - ergodicity; fractional brownian motion; Hörmander’s theorem; fractional Brownian motion
UR - http://eudml.org/doc/242429
ER -

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