  The addition of lower level integrality constraints to a bi-level linear
program is known to result in significantly weaker analytical properties. Most
notably, the upper level goal function in the optimistic setting lacks lower
semicontinuity and the existence of an optimal solution cannot be guaranteed
under standard assumptions. In this paper, we study a setting where the
right-hand side of the lower level constraint system is affected by the
leader's choice as well as the realization of some random vector. Assuming that
only the follower decides under complete information, we employ a convex risk
measure to assess the upper level outcome. Confining the analysis to the cases
where the lower level feasible set is finite, we provide sufficient conditions
for H\"older continuity of the leader's risk functional and draw conclusions
about the existence of optimal solutions. Finally, we examine qualitative
stability with respect to perturbations of the underlying probability measure.
Considering the topology of weak convergence, we prove joint continuity of the
objective function with respect to both the leader's decision and the
underlying probability measure.
