# Partial functional quantization and generalized bridges

Abstract : In this article, we develop a new approach to functional quantization, which consists in discretizing only a finite subset of the 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 a bigger filtration. This allows us 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 SDEs 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 SDEs, to using the Euler scheme of these SDEs in each Voronoi cell.
Keywords :
Document type :
Preprints, Working Papers, ...
2012
Domain :

https://hal.archives-ouvertes.fr/hal-00560275
Contributor : Sylvain Corlay <>
Submitted on : Wednesday, September 19, 2012 - 5:51:27 AM
Last modification on : Tuesday, October 11, 2016 - 1:55:30 PM
Document(s) archivé(s) le : Thursday, December 20, 2012 - 3:45:39 AM

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partial_quantization.pdf
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### Identifiers

• HAL Id : hal-00560275, version 4
• ARXIV : 1101.5488

### Citation

Sylvain Corlay. Partial functional quantization and generalized bridges. 2012. <hal-00560275v4>

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