# Absolute continuity and Fokker-Planck equation for the law of Wong-Zakai approximations of It\^o's stochastic differential equations

## TL;DR

This paper studies the regularity of the probability law of Wong-Zakai approximations for Itô SDEs, establishing absolute continuity and a Fokker-Planck equation using Malliavin calculus, despite the lack of parabolic smoothing.

## Contribution

It introduces a novel approach to prove absolute continuity for Wong-Zakai approximations by requiring the initial condition to have a density, compensating for missing smoothing effects.

## Key findings

- Established absolute continuity of the law of approximations.
- Derived a Fokker-Planck-type equation in the distributional sense.
- Identified the necessity of initial density for absolute continuity.

## Abstract

We investigate the regularity of the law of Wong-Zakai-type approximations for It\^o stochastic differential equations. These approximations solve random differential equations where the diffusion coefficient is Wick-multiplied by the smoothed white noise. Using a criteria based on the Malliavin calculus we establish absolute continuity and a Fokker-Planck-type equation solved in the distributional sense by the density. The parabolic smoothing effect typical of the solutions of It\^o equations is lacking in this approximated framework; therefore, in order to prove absolute continuity, the initial condition of the random differential equation needs to possess a density itself.