Partial functional quantization and generalized bridges

TL;DR

This paper introduces a novel partial functional quantization method for Gaussian semimartingales, enabling efficient approximation of SDE solutions by discretizing key coordinates and analyzing error bounds and convergence properties.

Contribution

It develops a new partial quantization approach using Karhunen-Loève coordinates and filtration enlargement, providing error bounds and convergence analysis for SDE solutions.

Findings

Upper bounds for $L^p$-partial quantization error of SDE solutions.

Conditional distribution of a Gaussian semimartingale given its quantization is a non-Gaussian semimartingale.

Investigation of almost sure convergence of the partial quantization process.

Abstract

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 $L^{p+\varepsilon}$-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 SDEs, to using the Euler scheme of these SDEs in each Voronoi cell.