  We study the singular stochastic optimal control problem with model
uncertainty, where the necessary conditions determined by the corresponding
maximum principle are trivial. Robust integral form and pointwise second order
necessary optimality conditions under certain compactness conditions are
derived. Both the drift and diffusion terms are control dependent but the
control region are assumed to be convex. The convex variational method is
employed, because linear structure is essential in deriving the weak limit of
uncertainty measures. Other main technical ingredients in obtaining the
integral type conditions are compact analysis and minimax theorem, while for
the pointwise ones it is Clark-Ocone formula and Lebesgue differentiation type
theorem. Besides, a compendious example is given to illustrate the motivation
and effectiveness of the results.
