Singular optimal controls for stochastic recursive systems under convex control constraint

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