  This paper studies duality and optimality conditions for general convex
stochastic optimization problems. The main result gives sufficient conditions
for the absence of a duality gap and the existence of dual solutions in a
locally convex space of random variables. It implies, in particular, the
necessity of scenario-wise optimality conditions that are behind many
fundamental results in operations research, stochastic optimal control and
financial mathematics. Our analysis builds on the theory of Fr\'echet spaces of
random variables whose topological dual can be identified with the direct sum
of another space of random variables and a space of singular functionals. The
results are illustrated by deriving sufficient and necessary optimality
conditions for several more specific problem classes. We obtain significant
extensions to earlier models e.g.\ on stochastic optimal control, portfolio
optimization and mathematical programming.
