Regularity of stochastic differential equations on the Wiener space by coupling

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 ${\text{D}}_{1,2}$ 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.