Convergence of multi-dimensional quantized $SDE$'s

TL;DR

This paper introduces a method for quantizing multidimensional Stratonovich SDEs using functional stationary quantizers, establishing convergence to the true SDE solution and providing quantization convergence rates.

Contribution

It connects quantization of Brownian motion with rough path theory to prove convergence of quantized SDE solutions, offering new insights and rates.

Findings

Quantized solutions converge to the SDE solution.

Provided convergence rates for optimal quantizations.

Established links between quantization and rough path theory.

Abstract

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(in the Stratonovich sense) by solving the related system of $ODE$'s in which the $d$-dimensional Brownian motion has been replaced by the components of functional stationary quantizers.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 $L^p(\P)$.