  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 optimal quadratic quantization
and the variance reduction that can be achieved with stratified sampling. We
first put the emphasis on the consistency of quantization for partitioning the
state space in stratified sampling methods in both finite and infinite
dimensional cases. We show that the proposed quantization-based strata design
has 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 such as Brownian bridge
or Ornstein-Uhlenbeck processes. We derive in detail the case of
Ornstein-Uhlenbeck processes. We also study the balance between the algorithmic
complexity of the simulation and the variance reduction factor
