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In this paper, we study two kinds of singular optimal controls (SOCs forshort) problems where the systems governed by forward-backward stochasticdifferential equations (FBSDEs for short), in which the control has twocomponents: the regular control, and the singular one. Both drift and diffusionterms may involve the regular control variable. The regular control domain ispostulated to be convex. Under certain assumptions, in the framework of theMalliavin calculus, we derive the pointwise second-order necessary conditionsfor stochastic SOC in the classical sense. This condition is described by twoadjoint processes, a maximum condition on the Hamiltonian supported by anillustrative example. A new necessary condition for optimal singular control isobtained as well. Besides, as a by-product, a verification theorem for SOCs isderived via viscosity solutions without involving any derivatives of the valuefunctions. It is worth pointing out that this theorem has wider applicabilitythan the restrictive classical verification theorems. Finally, we focus on theconnection between the maximum principle and the dynamic programming principlefor such SOCs problem without the assumption that the value function is smoothenough.

Singular Optimal Controls for Stochastic Recursive Systems under Convex Control Constraint