Epi-convergent discretizations of stochastic programs via integration quadratures

Modern integration quadratures are designed to produce finitely supported approximations of a given (probability) measure. This makes them well suited for discretization of stochastic programs. We give conditions that guarantee the epi-convergence of resulting objectives to the original one. Our epi-convergence result is closely related to some of the exisiting ones but it is easier to apply to discretizations. As examples, we will verify the conditions for discretizations of three different models of portfolio management and we study the behavior of various discretizations numerically. In our tests, modern quadratures clearly outperform crude Monte Carlo sampling in discretization of stochastic programs