Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60E99 None of the above, but in this section
- 60H10 Stochastic ordinary differential equations [See also 34F05]

**Résumé:** The functional quantization problem for
one-dimensional Brownian diffusions on $[0,T]$ is investigated.
First, the
existence of optimal
$n$-quantizers for every $n\ge 1$ is established, based on an
existence result for random vectors taking their values in abstract
Banach
spaces. Then, one shows under rather general assumptions that the
rate of convergence of the $L^p$-quantization error
is $O((\log n) ^{-\frac 12})$ like for the Brownian motion. Several
methods to construct some quasi-optimal quantizers are proposed.
Finally,
a special attention is given to diffusions with a Gaussian martingale term.

**Mots Clés:** *Functional quantization ; optimal quantizers ; Brownian diffusions ;
Lamperti transform ; Girsanov Theorem*

**Date:** 2003-10-15

**Prépublication numéro:** *PMA-853*