Smoothness of the density for solutions to Gaussian rough differential equations

TL;DR

This paper proves that solutions to certain Gaussian-driven stochastic differential equations have smooth probability densities, using advanced mathematical tools like rough path theory and Malliavin calculus, under specific nondegeneracy and Hörmander conditions.

Contribution

It establishes the smoothness of densities for solutions to Gaussian rough differential equations under Hörmander’s condition, extending previous results to a broad class of Gaussian processes.

Findings

Solutions have smooth densities for all positive times.

Applicable to fractional Brownian motion with Hurst parameter > 1/4.

Includes examples like Ornstein-Uhlenbeck process and Brownian bridge.

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 H\"{o}rmander'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 $T$.