Wong-Zakai approximation of density functions

Abstract

In this paper we prove the Wong-Zakai approximation of probability density functions of solutions at a fixed time of rough differential equations driven by fractional Brownian rough path with Hurst parameter $H$ $(1/4 <H \leq 1/2)$. Besides rough path theory, we use Hu-Watanabe's approximation theorem in the framework of Watanabe's distributional Malliavin calculus. When $H=1/2$, the random rough differential equations coincide with the corresponding Stratonovich-type stochastic differential equations. Even for that case, our main result seems new.