Densities for Rough Differential Equations under Hoermander's Condition
Probability
2007-08-29 v1
Abstract
We consider stochastic differential equations dY=V(Y)dX driven by a multidimensional Gaussian process X in the rough path sense. Using Malliavin Calculus we show that Y(t) admits a density for t in (0,T] provided (i) the vector fields V=(V_1,...,V_d) satisfy Hoermander'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.
Cite
@article{arxiv.0708.3730,
title = {Densities for Rough Differential Equations under Hoermander's Condition},
author = {Thomas Cass and Peter Friz},
journal= {arXiv preprint arXiv:0708.3730},
year = {2007}
}