Densities for rough differential equations under Hörmander's condition

Thomas Cass, Peter Friz
2010 Annals of Mathematics  
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 Hörmander's condition and (ii) the Gaussian driving signal X satisfies certain conditions. Examples of driving signals include fractional Brownian motion with Hurst
more » ... ter H > 1=4, the Brownian bridge returning to zero after time T and the Ornstein-Uhlenbeck process. PROPOSITION 1. Let x be a geometric p-rough path over ‫ޒ‬ d and h 2 C q-var OE0; T ; ‫ޒ‬ d such that 1=p C 1=q > 1. Then
doi:10.4007/annals.2010.171.2115 fatcat:b5aadp2dxndgpe7glkfkbiz7g4