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Smoothness of the density for solutions to Gaussian rough differential equations
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Cass, Thomas, Hairer, Martin, Litterer, Christian and Tindel, Samy (2015) Smoothness of the density for solutions to Gaussian rough differential equations. The Annals of Probability, 43 (1). pp. 188-239. doi:10.1214/13-AOP896 ISSN 0091-1798.
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Official URL: http://dx.doi.org/10.1214/13-AOP896
Abstract
We consider stochastic differential equations of the form dYt=V(Yt)dXt+V0(Yt)dt driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields V0 and V=(V1,…,Vd) satisfy Hörmander’s bracket condition, we demonstrate that Yt admits a smooth density for any t∈(0,T], provided the driving noise satisfies certain nondegeneracy assumptions. Our analysis relies on relies on an interplay of rough path theory, Malliavin calculus and the theory of Gaussian processes. Our result applies to a broad range of examples including fractional Brownian motion with Hurst parameter H>1/4, the Ornstein–Uhlenbeck process and the Brownian bridge returning after time T.
Item Type: | Journal Article | ||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Journal or Publication Title: | The Annals of Probability | ||||||
Publisher: | Institute of Mathematical Statistics | ||||||
ISSN: | 0091-1798 | ||||||
Official Date: | February 2015 | ||||||
Dates: |
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Volume: | 43 | ||||||
Number: | 1 | ||||||
Page Range: | pp. 188-239 | ||||||
DOI: | 10.1214/13-AOP896 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Restricted or Subscription Access |
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