  This paper investigates the use of stratified sampling as a variance
reduction technique for approximating integrals over large dimensional spaces.
The accuracy of this method critically depends on the choice of the space
partition, the strata, which should be ideally fitted to thesubsets where the
functions to integrate is nearly constant, and on the allocation of the number
of samples within each strata. When the dimension is large and the function to
integrate is complex, finding such partitions and allocating the sample is a
highly non-trivial problem. In this work, we investigate a novel method to
improve the efficiency of the estimator "on the fly", by jointly sampling and
adapting the strata and the allocation within the strata. The accuracy of
estimators when this method is used is examined in detail, in the so-called
asymptotic regime (i.e. when both the number of samples and the number of
strata are large). We illustrate the use of the method for the computation of
the price of path-dependent options in models with both constant and stochastic
volatility. The use of this adaptive technique yields variance reduction by
factors sometimes larger than 1000 compared to classical Monte Carlo
estimators.
