Given an arbitrary number of risk-averse or risk-neutral convex stochastic programs, we study hypotheses testing problems aiming at comparing the optimal values of these stochastic programs on the basis of samples of the underlying random vectors. We propose non-asymptotic tests based on confidence intervals on the optimal values of the stochastic programs obtained using the Robust Stochastic Approximation and the Stochastic Mirror Descent algorithms. When the objective functions are uniformly convex, we also propose a multi-step version of the Stochastic Mirror Descent algorithm and obtain confidence intervals on both the optimal values and optimal solutions. The results are applied to compare, using tests of hypotheses, the (extended polyhedral) risk measure values of several distributions.