Due to their frequently observed lack of convexity and/or smoothness, stochastic programs with joint probabilistic constraints have been considered as a hard type of constrained optimization problems, which are rather demanding both from the computational and robustness point of view. Dependence of the set of solutions on the probability distribution rules out the straightforward construction of the convexity-based global contamination bounds for the optimal value; at least local results for probabilistic programs of a special structure will be derived. Several alternative approaches to output analysis will be mentioned.