  Optimality conditions in the form of a variational inequality are proved for
a class of constrained optimal control problems of stochastic differential
equations. The cost function and the inequality constraints are functions of
the probability distribution of the state variable at the final time. The
analysis uses in an essential manner a convexity property of the set of
reachable probability distributions. An augmented Lagrangian method based on
the obtained optimality conditions is proposed and analyzed for solving
iteratively the problem. At each iteration of the method, a standard stochastic
optimal control problem is solved by dynamic programming. Two academical
examples are investigated.
