Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Functional quantization of $1$-dimensional Brownian diffusion processes

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