Convergence of multi-dimensional quantized $SDE$s

We quantize a multidimensional $SDE$ (in the Stratonovich sense) by solving the related system of $ODE$s in which the $d$-dimensional Brownian motion has been replaced by the components of functional stationary quantizers. We make a connection with rough path theory to show that the solutions of the quantized solutions of the $ODE$ converge toward the solution of the $SDE$. On our way to this result we provide convergence rates of optimal quantizations toward the Brownian motion for $frac 1q$-H older distance, $q>2$, in $L^p(P)$.