Regularity of stochastic differential equations on the Wiener space by coupling

TL;DR

This paper explores the regularity of scalar stochastic differential equations on Wiener space using coupling methods, focusing on Lipschitz and Hölder continuous diffusion coefficients, and examines their implications for Sobolev spaces and backward SDEs.

Contribution

It introduces coupling techniques to analyze regularity properties of SDEs with non-smooth coefficients, extending existing methods to new functional spaces and backward equations.

Findings

Coupling methods effectively analyze regularity in Sobolev and interpolation spaces.

The approach handles Hölder continuous diffusion coefficients.

Results provide new insights into the variation of backward SDEs.

Abstract

Using the coupling method introduced in \cite{Geiss:Ylinen:21}, we investigate regularity properties of stochastic differential equations, where we consider the Lipschitz case in $\R^d$ and allow for H\"older continuity of the diffusion coefficient of scalar valued stochastic differential equations. Two cases of the coupling method are of special interest: The uniform coupling to treat the Malliavin Sobolev space $\D_{1,2}$ and real interpolation spaces, and secondly a cut-off coupling to treat the $L_p$-variation of backward stochastic differential equations where the forward process is the investigated stochastic differential equation.