Adaptive optimal allocation in stratified sampling methods

In this paper, we propose a stratified sampling algorithm in which the random drawings made in the strata to compute the expectation of interest are also used to adaptively modify the proportion of further drawings in each stratum. These proportions converge to the optimal allocation in terms of variance reduction. And our stratified estimator is asymptotically normal with asymptotic variance equal to the minimal one. Numerical experiments confirm the efficiency of our algorithm. Introduction Let X be a R-valued random variable and f : R → R a measurable function such that E(f(X)) < ∞. We are interested in the computation of c = E(f(X)) using a stratified sampling Monte-Carlo estimator. We suppose that (Ai)1≤i≤I is a partition of R into I strata such that pi = P[X ∈ Ai] is known explicitely for i ∈ {1, . . . , I}. Up to removing some strata, we assume from now on that pi is positive for all i ∈ {1, . . . , I}. The stratified Monte-Carlo estimator of c (see [G04] p.209-235 and the references therein for a presentation more detailed than the current introduction) is based on the equality E(f(X)) = ∑I i=1 piE(f(Xi)) whereXi denotes a random variable distributed according to the conditional law of X given X ∈ Ai. Indeed, when the variables Xi are simulable, it is possible to estimate each expectation in the right-hand-side using Ni i.i.d drawings of Xi. Let N = ∑I i=1 Ni be the total number of drawings (in all the strata) and qi = Ni/N denote the proportion of drawings made in stratum i. Then ĉ is defined by