  We consider pessimistic bilevel stochastic programs in which the follower
maximizes over a fixed compact convex set a strictly convex quadratic function,
whose Hessian depends on the leader's decision. The resulting random variable
is evaluated by a convex risk measure. Under assumptions including real
analyticity of the lower-level goal function, we prove existence of optimal
solutions. We discuss an alternate model where the leader hedges against
optimal lower-level solutions, and show that in this case solvability can be
guaranteed under weaker conditions both in a deterministic and in a stochastic
setting. The approach is applied to a mechanical shape optimization problem in
which the leader decides on an optimal material distribution to minimize a
tracking-type cost functional, whereas the follower chooses forces from an
admissible set to maximize a compliance objective. The material distribution is
considered to be stochastically perturbed in the actual construction phase.
Computational results illustrate the bilevel optimization concept and
demonstrate the interplay of follower and leader in shape design and testing.
