Partial functional quantization and generalized bridges

Sylvain Corlay (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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 $L^p$-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.

Keywords: Gaussian semimartingale; functional quantization; vector quantization; Karhunen-Loève; Gaussian process; Brownian motion; Brownian bridge; Ornstein-Uhlenbeck; filtration enlargement; stratification; Cameron-Martin space; Wiener integral

Domains: Probability [math.PR]

Sylvain Corlay. Partial functional quantization and generalized bridges. 2012. ⟨hal-00560275v4⟩