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  . Using filtration enlargement techniques, we prove that the conditional distribution of 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 by simply plugging the partial functional quantization of in the SDE.
Then, we provide an upper bound of the  -partial quantization error for the solution of SDE involving the  -partial quantization error for , for  . The  convergence is also investigated.
Incidentally, we show that the conditional distribution of a Gaussian semimartingale 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.
References- Sylvain Corlay, and Gilles Pagès,
Functional quantization based stratified sampling methods,
, 2010.
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