@article {87,
title = {Partial functional quantization and generalized bridges},
year = {2011},
abstract = {In this article, we develop a new approach to functional quantization, which consists in discretizing only the first Karhunen-Lo{\`e}ve coordinates of a continuous Gaussian semimartingale $X$. Using filtration enlargement techniques, we prove that the conditional distribution of $X$ knowing its first Karhunen-Lo{\`e}ve 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 [bib]CorlayPagesStratification[/bib], amounted in the case of solutions of SDE to use the Euler scheme of these SDE in each Voronoi cell.},
keywords = {Brownian bridge, Brownian motion, filtration enlargement, functional quantization, Gaussian process, Gaussian semimartingale, Karhunen-Lo{\`e}ve, Ornstein-Uhlenbeck, stratification, vector quantization},
attachments = {http://www.quantize.maths-fi.com/sites/default/files/partial quantization.pdf},
author = {Sylvain Corlay}
}