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Article Dans Une Revue Annals of Probability Année : 2015

Smoothness of the density for solutions to Gaussian rough differential equations

Résumé

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.
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Dates et versions

hal-00732063 , version 1 (13-09-2012)

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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⟩
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