Smoothness of the density for solutions to Gaussian rough differential equations

Citation
Thomas, Cass et al., Smoothness of the density for solutions to Gaussian rough differential equations, Annals of probability (Online) , 43(1), 2015, pp. 188-239
ISSN journal
2168894X
Volume
43
Issue
1
Year of publication
2015
Pages
188 - 239
Database
ACNP
SICI code
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.