Robust pointwise second order necessary conditions for singular stochastic optimal control with model uncertainty

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.