Diierentiable Selections of Set-valued Mappings with Application in Stochastic Programming

We consider set-valued mappings deened on a linear normed space with convex closed images in IR n. Our aim is to construct selections which are (Hadamard) directionally diierentiable using some approximation of the mul-tifunction. The constructions suggested assume existence of a cone approximation given by a certain "derivative" of the mapping. The rst one makes use of the properties of Steiner points. The notion of Steiner center is generalized for a class of unbounded sets, which include the polyhedral sets. The second construction deenes a continuous selection through a given point of the graph of the multifunction and being Hadamard directionally diierentiable at that point with derivatives belonging to the corresponding "derivative" of the multifunction. Both constructions lead to a directionally diierentiable Castaing representation of measurable multifunctions with the required diierentiability properties. The results are applied to obtain statements about the asymptotic behaviour of measurable selections of random sets via the delta-approach. Particularly, random sets of this kind build the solutions of two-stage stochastic programs.