Regularity of stochastic differential equations on the Wiener space by coupling

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 ${\mathbb{R}}^d$ and allow for Hölder 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.

arXiv commentResults added, improvement of presentation, and change of title to reflect better the content of the manuscript