Densities for Rough Differential Equations under Hoermander's Condition

TL;DR

This paper proves the existence of probability densities for solutions to rough differential equations driven by Gaussian processes, under Hörmander's condition and specific properties of the driving signals, extending classical results to rough paths.

Contribution

It establishes conditions under which solutions to rough differential equations driven by Gaussian signals have smooth densities, using Malliavin Calculus and Hörmander's condition.

Findings

Density exists for fractional Brownian motion with H>1/4

Density exists for Ornstein-Uhlenbeck process

Density exists for Brownian Bridge returning to zero

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.