  We revisit the sample average approximation (SAA) approach for non-convex
stochastic programming. We show that applying the SAA approach to problems with
expected value equality constraints does not necessarily result in asymptotic
optimality guarantees as the sample size increases. To address this issue, we
relax the equality constraints. Then, we prove the asymptotic optimality of the
modified SAA approach under mild smoothness and boundedness conditions on the
equality constraint functions. Our analysis uses random set theory and
concentration inequalities to characterize the approximation error from the
sampling procedure. We apply our approach and analysis to the problem of
stochastic optimal control for nonlinear dynamical systems under external
disturbances modeled by a Wiener process. Numerical results on relevant
stochastic programs show the reliability of the proposed approach. Results on a
rocket-powered descent problem show that our computed solutions allow for
significant uncertainty reduction compared to a deterministic baseline.
