21787 articles – 15600 Notices  [english version]
 HAL : hal-00560275, version 1
 arXiv : 1101.5488
 Versions disponibles : v1 (28-01-2011) v2 (02-03-2011) v3 (27-04-2011) v4 (19-09-2012)
 Partial functional quantization and generalized bridges
 (27/01/2011)
 In this article, we develop a new approach to functional quantization, which consists in discretizing only the first Karhunen-Loève coordinates of a continuous Gaussian semimartingale $X$. Using filtration enlargement techniques, we prove that the conditional distribution of $X$ knowing its first Karhunen-Loève coordinates is a Gaussian semimartingale with respect to its natural filtration. This allows to define the partial quantization of a solution of a stochastic differential equation with respect to $X$ by simply plugging the partial functional quantization of $X$ in the SDE. Then, we provide an upper bound of the $L^p$-partial quantization error for the solution of SDE involving the $L^{p+\varepsilon}$-partial quantization error for $X$, for $\varepsilon >0$. The $a.s.$ convergence is also investigated. Incidentally, we show that the conditional distribution of a Gaussian semimartingale $X$ knowing that it stands in some given Voronoi cell of its functional quantization is a (non-Gaussian) semimartingale. As a consequence, the functional stratification method developed in [6], amounted in the case of solutions of SDE to simulate use the Euler scheme of these SDE in each Voronoi cell.
 1 : Laboratoire de Probabilités et Modèles Aléatoires (LPMA) CNRS : UMR7599 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot
 Domaine : Mathématiques/Probabilités
 Mots Clés : Gaussian semimartingale – functional quantization – vector quantization – Karhunen-Loève – Gaussian process – Brownian motion – Brownian bridge – Ornstein-Uhlenbeck – filtration enlargement – stratification
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 hal-00560275, version 1 http://hal.archives-ouvertes.fr/hal-00560275 oai:hal.archives-ouvertes.fr:hal-00560275 Contributeur : Sylvain Corlay <> Soumis le : Jeudi 27 Janvier 2011, 18:54:05 Dernière modification le : Vendredi 28 Janvier 2011, 10:38:51