  The vast majority of the literature on stochastic semidefinite programs
(stochastic SDPs) with recourse is concerned with risk-neutral models. In this
paper, we introduce mean-risk models for stochastic SDPs and study structural
properties as convexity and (Lipschitz) continuity. Special emphasis is placed
on stability with respect to changes of the underlying probability
distribution. Perturbations of the true distribution may arise from incomplete
information or working with (finite discrete) approximations for the sake of
computational efficiency. We discuss extended formulations for stochastic SDPs
under finite discrete distributions, which turn out to be deterministic
(mixed-integer) SDPs that are (almost) block-structured for many popular risk
measures.
