Densities for rough differential equations under Hörmander’s condition

Abstract

We consider stochastic differential equations d Y D V .Y / 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 2 .0; T provided (i) the vector fields V D .V 1 ; : : : ; V d / satisfy Hrmander'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.