Densities for rough differential equations under Hörmander's condition

Abstract

We consider stochastic differential equations dY = 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 Yt admits a density for t ∈ (0,T] provided (i) the vector fields V = (V1, . . ., Vd) 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. © 2010 Annals of Mathematics.