Partial functional quantization and generalized bridges
Résumé
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 [6], amounted in the case of solutions of SDE to simulate use the Euler scheme of these SDE in each Voronoi cell.
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