Convergence of multi-dimensional quantized $SDE$'s

Details Convergence of multi-dimensional quantized $SDE$'s Convergence of multi-dimensional quantized $SDE$'s

2008-01-04

arxiv.org/abs/0801.0726v2

Mathematics

Probability

43 pages

Scientific paper

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(in the Stratonovich sense) by solving the related system of $ODE$'s in which the $d$-dimensional Brownian motion has been replaced by the components of functional stationary quantizers.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

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