Quantitative Stability Analysis of Stochastic Generalized Equations

Abstract

We consider the solution of a system of stochastic generalized equations (SGE) where theunderlying functions are mathematical expectation of random set-valued mappings. SGE hasmany applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem and a stochastic equilibrium problem. We derive quantitative continuityof expected value of the set-valued mapping with respect to the variation of the underlyingprobability measure in a metric space. This leads to the subsequent qualitative and quantitative stability analysis of solution set mappings of the SGE. Under some metric regularityconditions, we derive Aubin’s property of the solution set mapping with respect to the changeof probability measure. The established results are applied to stability analysis of stationary points of classical one stage and two stage stochastic minimization problems, two stagestochastic mathematical programs with equilibrium constraints and stochastic programs withsecond order dominance constraints.