  We consider bilevel linear problems, where some parameters are stochastic,
and the leader has to decide in a here-and-now fashion, while the follower has
complete information. In this setting, the leader's outcome can be modeled by a
random variable, which we evaluate based on some law-invariant convex risk
measure. A qualitative stability result under perturbations of the underlying
probability distribution is presented. Moreover, for the expectation, the
expected excess, and the upper semideviation, we establish Lipschitz continuity
as well as sufficient conditions for differentiability. Finally, for finite
discrete distributions, we reformulate the bilevel stochastic problems as
standard bilevel problems and propose a regularization scheme for bilevel
linear problems.
