Asymptotic control of the mean-squared error for Monte Carlo sensitivity index estimators in stochastic models

In global sensitivity analysis for stochastic models, the Sobol' sensitivity index is a ratio of polynomials in which each variable is an expectation of a function of a conditional expectation. The estimator is then based on nested Monte Carlo sampling where the sizes of the inner and outer loops correspond to the number of repetitions and explorations, respectively. Under some conditions, it was shown that the optimal rate of the mean squared error for estimating the expectation of a function of a conditional expectation by nested Monte Carlo sampling is of order the computational budget raised to the power-2/3. However, the control of the mean squared error for ratios of polynomials is more challenging. We show the convergence in quadratic mean of the Sobol' index estimator. A bound is found that allows us to propose an allocation strategy based on a bias-variance trade-off. A practical algorithm that adapts to the model intrinsic randomness and exploits the knowledge of the optimal allocation is proposed and illustrated on numerical experiments.