Partial functional quantization and generalized bridges

TitlePartial functional quantization and generalized bridges
Publication TypeJournal Article
Year of Publication2011
AuthorsSylvain Corlay
KeywordsBrownian bridge, Brownian motion, filtration enlargement, functional quantization, Gaussian process, Gaussian semimartingale, Karhunen-Loève, Ornstein-Uhlenbeck, stratification, vector quantization

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 [1], amounted in the case of solutions of SDE to use the Euler scheme of these SDE in each Voronoi cell.


  1. Sylvain Corlay, and Gilles Pagès, Functional quantization based stratified sampling methods, , 2010.
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