In this article, we propose several quantization based stratified sampling
methods to reduce the variance of a Monte-Carlo simulation. Theoretical aspects
of stratification lead to a strong link between the problem of optimal
$L^2$-quantization of a random variable and the variance reduction that can be
achieved. We first emphasize on the consistency of quantization for designing
strata in stratified sampling methods in both finite dimensional and infinite
dimensional frameworks. We show that this strata design has a uniform
efficiency among the class of Lipschitz continuous functionals. Then a
stratified sampling algorithm based on product functional quantization is
proposed for path-dependent functionals of multi-factor diffusions. The method
is also available for other Gaussian processes as the Brownian bridge or an
Ornstein-Uhlenbeck process. We derive in detail the quantization of the
Ornstein-Uhlenbeck process. The balance between the algorithmic complexity of
the simulation and the variance reduction factor has also been studied.