Annals of Probability

Smoothness of the density for solutions to Gaussian rough differential equations

Thomas Cass, Martin Hairer, Christian Litterer, and Samy Tindel

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We consider stochastic differential equations of the form $dY_{t}=V(Y_{t})\,dX_{t}+V_{0}(Y_{t})\,dt$ driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields $V_{0}$ and $V=(V_{1},\ldots,V_{d})$ satisfy Hörmander’s bracket condition, we demonstrate that $Y_{t}$ admits a smooth density for any $t\in(0,T]$, 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 $T$.

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Ann. Probab., Volume 43, Number 1 (2015), 188-239.

First available in Project Euclid: 12 November 2014

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Zentralblatt MATH identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60G15: Gaussian processes 60H10: Stochastic ordinary differential equations [See also 34F05]

Rough path analysis Gaussian processes Malliavin calculus


Cass, Thomas; Hairer, Martin; Litterer, Christian; Tindel, Samy. Smoothness of the density for solutions to Gaussian rough differential equations. Ann. Probab. 43 (2015), no. 1, 188--239. doi:10.1214/13-AOP896.

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