Existence of density for solutions of mixed stochastic equations

TL;DR

This paper proves that solutions to certain mixed stochastic differential equations driven by Wiener and fractional Brownian motions have a density, under Hormander conditions, extending the understanding of their probabilistic properties.

Contribution

It establishes the existence of a density for solutions of mixed stochastic equations driven by independent Wiener and fractional Brownian motions under Hormander conditions.

Findings

Solutions have a density with respect to Lebesgue measure.

Density existence is proven under Hormander type conditions.

Extends results to mixed stochastic differential equations.

Abstract

We consider a mixed stochastic differential equation $d{X_t}=a(t,X_t)d{t}+b(t,X_t) d{W_t}+c(t,X_t)d{B^H_t}$ driven by independent multidimensional Wiener process and fractional Brownian motion. Under Hormander type conditions we show that the distribution of $X_t$ possesses a density with respect to the Lebesgue measure.