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-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-partial quantization error for the solution of SDEs involving the L-partial quantization error for X, for ε > 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.