Di erentiable Selections of Set Valued Mappings With Application in Stochastic Programming

We consider set valued mappings de ned on a linear normed space with convex closed images in IRn Our aim is to construct selections which are Hadamard directionally di erentiable using some approximation of the mul tifunction The constructions suggested assume existence of a cone approxima tion 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 de nes a continuous selection through a given point of the graph of the multifunction and being Hadamard directionally di erentiable at that point with derivatives belonging to the corresponding derivative of the multifunction Both constructions lead to a directionally di erentiable Castaing representation of measurable multifunctions with the required di erentiability properties The results are applied to obtain statements about the asymptotic behaviour of mea surable selections of random sets via the delta approach Particularly random sets of this kind build the solutions of two stage stochastic programs