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Convergence of multi-dimensional quantized $SDE$'s

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(in the Stratonovich sense) by solving the related system of $ODE$'s in which the $d$-dimensional Brownian motion has been replaced by the components of functional stationary quantizers.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 $L^p(\P)$.
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https://hal.archives-ouvertes.fr/hal-00202297
Contributor : Gilles Pagès <>
Submitted on : Monday, July 5, 2010 - 9:56:36 PM
Last modification on : Wednesday, December 9, 2020 - 3:09:17 PM
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Gilles Pagès, Afef Sellami. Convergence of multi-dimensional quantized $SDE$'s. Séminaire de Probabilités, Springer-Verlag, 2011, 2006 (Lecture Notes in Math.), 269-307 ; http://dx.doi.org/10.1007/978-3-642-15217-7_11. ⟨10.1007/978-3-642-15217-7_11⟩. ⟨hal-00202297v2⟩

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