Minimization of stochastic functionals and random variational inequalities

Abstract

We examine the problem of minimizing a stochastic functional on a closed convex subset of a Hilbert space and provide sufficient conditions for the existence of a solution. As a departure from the literature, none of these conditions is imposed on all sample paths. To argue for the need to avoid such conditions, we prove that the Ito integral on L 2(0, 1) possesses neither l.s.c. nor convex sample paths. The theory is utilized to verify the existence of a minimum for a natural extension of a functional determined by randomly perturbing the energy in the 1 dimensional obstacle problem. In the second part of the paper we establish a link between random variational inequalities and minimization of stochastic functional, the link is employed to establish the existence of solutions for some cases of the former