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Convergence of Multi-Dimensional Quantized SDE’s

  • Gilles Pagès
  • Afef Sellami
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2006)

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SDE (in the Stratonovich sense) by solving the related system of ODE’s in which the d-dimensional Brownian motion has been replaced by the components of functional stationary quantizers.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 \(\frac{1} {q}\)-Hölder distance, q > 2, in L p ().

Functional quantization Stochastic differential equations Stratonovich stochastic integral Stationary quantizers Rough path theory Itô map Hölder semi-norm p-variation 

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Notes

…cknowledgements

We thank A. Lejay for helpful discussions and comments about several versions of this work and F. Delarue and S. Menozzi for initiating our first “meeting” with rough path theory.

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Laboratoire de Probabilités et Modèles AléatoiresUniversité de Paris 6Paris Cedex 5France
  2. 2.JP Morgan, London & Laboratoire de Probabilités et Modèles AléatoiresUniversité de Paris 6Paris Cedex 5France

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