  In this paper, we study two kinds of singular optimal controls (SOCs for
short) problems where the systems governed by forward-backward stochastic
differential equations (FBSDEs for short), in which the control has two
components: the regular control, and the singular one. Both drift and diffusion
terms may involve the regular control variable. The regular control domain is
postulated to be convex. Under certain assumptions, in the framework of the
Malliavin calculus, we derive the pointwise second-order necessary conditions
for stochastic SOC in the classical sense. This condition is described by two
adjoint processes, a maximum condition on the Hamiltonian supported by an
illustrative example. A new necessary condition for optimal singular control is
obtained as well. Besides, as a by-product, a verification theorem for SOCs is
derived via viscosity solutions without involving any derivatives of the value
functions. It is worth pointing out that this theorem has wider applicability
than the restrictive classical verification theorems. Finally, we focus on the
connection between the maximum principle and the dynamic programming principle
for such SOCs problem without the assumption that the value function is smooth
enough.
