Densities for Rough Differential Equations under Hoermander's Condition

2007/08/28 by Cass, Thomas, Friz, Peter · 1 citation
#60G17 #60H07 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0708.3730

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=(V₁,...,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.

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