  Variational inequality problems allow for capturing an expansive class of
problems, including convex optimization problems, convex Nash games and
economic equilibrium problems, amongst others. Yet in most practical settings,
such problems are complicated by uncertainty, motivating the examination of a
stochastic generalization of the variational inequality problem and its
extensions in which the components of the mapping contain expectations. When
the associated sets are unbounded, ascertaining existence requires having
access to analytical forms of the expectations. Naturally, in practical
settings, such expressions are often difficult to derive, severely limiting the
applicability of such an approach. Consequently, our goal lies in developing
techniques that obviate the need for integration and our emphasis lies in
developing tractable and verifiable sufficiency conditions for claiming
existence. We begin by recapping almost-sure sufficiency conditions for
stochastic variational inequality problems with single-valued maps provided in
our prior work [44] and provide extensions to multi-valued mappings. Next, we
extend these statements to quasi-variational regimes where maps can be either
single or set-valued. Finally, we refine the obtained results to accommodate
stochastic complementarity problems where the maps are either general or
co-coercive. The applicability of our results is demonstrated on practically
occurring instances of stochastic quasi-variational inequality problems and
stochastic complementarity problems, arising as nonsmooth generalized
Nash-Cournot games and power markets, respectively.
