Partial functional quantization and generalized bridges

In this article, we develop a new approach to functional quantization, which consists in discretizing only a finite subset of the Karhunen-Lo`eve coordinates of a continuous Gaussian semimartingale $X$. Using filtration enlargement techniques, we prove that the conditional distribution of $X$ knowing its first Karhunen-Lo`eve 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 $L^{p+\epsilon }$-partial quantization error for $X$, for $\epsilon >0$. The $a\mathrm{.}s\mathrm{.}$ 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.