Smoothness of the density for solutions to Gaussian rough differential equations

• Cass, Thomas ^[1] ; Hairer, Martin ^[2] ; Litterer, Christian ^[1] ; Tindel, Samy ^[3]
  1. [1] Imperial College London

     Imperial College London

     Reino Unido

     
  2. [2] University of Warwick

     University of Warwick

     Reino Unido

     
  3. [3] University of Lorraine

     University of Lorraine

     Arrondissement de Nancy, Francia

     

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 1, 2015, págs. 188-239
• Idioma: inglés
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  • 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.