## The Annals of Probability

### Smoothness of the density for solutions to Gaussian rough differential equations

#### Abstract

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 $T$.

#### Article information

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

Dates
First available in Project Euclid: 12 November 2014

https://projecteuclid.org/euclid.aop/1415801556

Digital Object Identifier
doi:10.1214/13-AOP896

Mathematical Reviews number (MathSciNet)
MR3298472

Zentralblatt MATH identifier
1309.60055

#### Citation

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. https://projecteuclid.org/euclid.aop/1415801556

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