  In this paper, we consider optimal control of stochastic differential
equations subject to an expected path constraint. The stochastic maximum
principle is given for a general optimal stochastic control in terms of
constrained FBSDEs. In particular, the compensated process in our adjoint
equation is deterministic, which seems to be new in the literature. For the
typical case of linear stochastic systems and quadratic cost functionals (i.e.,
the so-called LQ optimal stochastic control), a verification theorem is
established, and the existence and uniqueness of the constrained reflected
FBSDEs are also given.
