  We analyze a convex stochastic optimization problem where the state is
assumed to belong to the Bochner space of essentially bounded random variables
with images in a reflexive and separable Banach space. For this problem, we
obtain optimality conditions that are, with an appropriate model, necessary and
sufficient. Additionally, the Lagrange multipliers associated with optimality
conditions are integrable vector-valued functions and not only measures. A
model problem is given demonstrating the application to PDE-constrained
optimization under uncertainty with an outlook for further applications.
