This paper is in review.Stochastic ambiguity provides a rich class of uncertainty models that includes those in
stochastic, robust, risk-based, and semi-in nite optimization, and that accounts for both uncertainty
about parameter values as well as incompleteness of the description of uncertainty. We provide a novel,
unifying perspective on optimization under stochastic ambiguity that rests on two pillars. First, the
paper models ambiguity by decision-dependent collections of cumulative distribution functions viewed
as subsets of a metric space of upper semicontinuous functions. We derive a series of results for this set-
ting including estimates of the metric, the hypo-distance, and a new proof of the equivalence with weak
convergence. Second, we utilize the theory of lopsided convergence to establish existence, convergence,
and approximation of solutions of optimization problems with stochastic ambiguity. For the rst time,
we estimate the lop-distance between bifunctions and show that this leads to bounds on the solution
quality for problems with stochastic ambiguity. Among other consequences, these results facilitate the
study of the \price of robustness" and related quantities
