Smoothness of the density for solutions to Gaussian rough differential equations

Thomas Cass (1) , Martin Hairer (2) , Christian Litterer (1) , Samy Tindel (3, 4)

1 Department of Mathematics [Imperial College London]
2 WMI - Warwick Mathematics Institute
3 BIGS - Biology, genetics and statistics
4 IECL - Institut Élie Cartan de Lorraine

We consider stochastic differential equations driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields satisfy Hörmander's bracket condition, we demonstrate that the solution admits a smooth density for any strictly positive time t, provided the driving noise satisfies certain non-degeneracy assumptions. Our analysis 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 greater than 1/4, the Ornstein-Uhlenbeck process and the Brownian bridge returning after time T.

Domaines: Probabilités [math.PR]

Thomas Cass, Martin Hairer, Christian Litterer, Samy Tindel. Smoothness of the density for solutions to Gaussian rough differential equations. Annals of Probability, 2015, 43 (1), pp.188-239. ⟨10.1214/13-AOP896⟩. ⟨hal-00732063⟩