  We consider a stochastic control problem where the set of strict (classical)
controls is not necessarily convex and the the variable control has two
components, the first being absolutely continuous and the second singular. The
system is governed by a nonlinear stochastic differential equation, in which
the absolutely continuous component of the control enters both the drift and
the diffusion coefficients. By introducing a new approach, we establish
necessary and sufficient optimality conditions for two models. The first
concerns the relaxed-singular controls, who are a pair of processes whose first
component is a measure-valued processes. The second is a particular case of the
first and relates to strict-singular control problems. These results are given
in the form of global stochastic maximum principle by using only the first
order expansion and the associated adjoint equation. This improves and
generalizes all the previous works on the maximum principle of controlled
stochastic differential equations.
