  Sobol' sensitivity index estimators for stochastic models are functions of
nested Monte Carlo estimators, which are estimators built from two nested Monte
Carlo loops. The outer loop explores the input space and, for each of the
explorations, the inner loop repeats model runs to estimate conditional
expectations. Although the optimal allocation between explorations and
repetitions of one's computational budget is well-known for nested Monte Carlo
estimators, it is less clear how to deal with functions of nested Monte Carlo
estimators, especially when those functions have unbounded Hessian matrices, as
it is the case for Sobol' index estimators. To address this problem, a
regularization method is introduced to bound the mean squared error of
functions of nested Monte Carlo estimators. Based on a heuristic, an allocation
strategy that seeks to minimize a bias-variance trade-off is proposed. The
method is applied to Sobol' index estimators for stochastic models. A practical
algorithm that adapts to the level of intrinsic randomness in the models is
given and illustrated on numerical experiments.
