Stability and Sensitivity of Stochastic Dominance Constrained Optimization Models

We consider convex optimization problems with kth order stochastic dominance constraints for k ≥ 2. We discuss distances of random variables that are relevant for the dominance relation and establish quantitative stability results for optimal values and solution sets of the optimization problems in terms of a suitably selected probability metric. Moreover, we provide conditions ensuring Hadamard directional differentiability of the optimal value function. We introduce the notion of a shadow utility, which determines the changes of the optimal value when the underlying random variables are perturbed. Finally, we derive a limit theorem for the optimal values of empirical (Monte Carlo, sample average) approximations of dominance constrained optimization models.