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February 2015 Smoothness of the density for solutions to Gaussian rough differential equations
Thomas Cass, Martin Hairer, Christian Litterer, Samy Tindel
Ann. Probab. 43(1): 188-239 (February 2015). DOI: 10.1214/13-AOP896


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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Thomas Cass. Martin Hairer. Christian Litterer. Samy Tindel. "Smoothness of the density for solutions to Gaussian rough differential equations." Ann. Probab. 43 (1) 188 - 239, February 2015.


Published: February 2015
First available in Project Euclid: 12 November 2014

zbMATH: 1309.60055
MathSciNet: MR3298472
Digital Object Identifier: 10.1214/13-AOP896

Primary: 60G15 , 60H07 , 60H10

Keywords: Gaussian processes , Malliavin calculus , rough path analysis

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 1 • February 2015
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