Smoothness of the density for solutions to Gaussian rough differential equations

By T. Cass, Martin Hairer, Christian Litterer, Samy Tindel. 2012

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We consider stochastic differential equations driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields satisfy Hormander'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.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/94b571f0be21f3af4e5bdcfb67f57405c29fcd4d