Asymptotics of solutions of the sample average approximation method to solve risk averse stochastic programs

Abstract

The paper studies the Sample Average Approximation method to solve risk averse stochastic programs expressed in terms of divergence risk measures. It continues the recent contribution [18] on the first order asymptotics of the optimal values. We study asymptotic statistical properties of the optimal solutions, deriving general convergence rates and asymptotic distributions. As a crucial point the investigations are based on a new type of conditions from the theory of empirical processes which do not rely on pathwise analytical properties of the goal functions. In particular, continuity in the parameter is not imposed in advance as usual in the literature on the Sample Average Approximation method. It is also shown that the new condition is satisfied if the paths of the goal functions are Hölder continuous so that the main results carry over in this case. Moreover, the main results are applied to goal functions whose paths are piecewise linear as e.g. in two stage mixed-integer programs.