Approximations of Stochastic Optimization Problems Subject to Measurability Constraints

Motivated by the numerical resolution of stochastic optimization problems subject to measurability constraints, we focus upon the issue of how to discretize the components arising in the problem formulation. By means of a counterexample based on Monte Carlo approximation, we emphasize the importance of independent discretization of, on the one side, the random variable modelling uncertainties (noise) and, on the other side, the σ-field modelling the knowledge (information). Then, we present conditions under which the discretized problems converge to the original one. The focus is put on the probabilistic convergence notions ensuring the convergence.