Densities for rough differential equations under Hörmander’s condition

Abstract

We consider stochastic differential equations $dY=V\left( Y\right) dX$ driven by a multidimensional Gaussian process $X$ in the rough path sense [T. Lyons, Rev. Mat. Iberoamericana 14, (1998), 215–310]. Using Malliavin Calculus we show that $Y_{t}$ admits a density for $t\in (0,T]$ provided (i) the vector fields $V=\left( V_{1},\dots,V_{d}\right) $ satisfy Hörmander’s condition and (ii) the Gaussian driving signal $X$ satisfies certain conditions. Examples of driving signals include fractional Brownian motion with Hurst parameter $H>1/4$, the Brownian bridge returning to zero after time $T$ and the Ornstein-Uhlenbeck process.