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 $T$.