  In this paper, we consider optimal control problems derived by stochastic
systems with delay, where control domains are non-convex and the diffusion
coefficients depend on control variables. By an estimate of the integral of
x_{1}(t)x_{1}(t-\delta) term, we obtain a general maximum principle for the
optimal control problems with a standard spike variational technique and
duality method. The maximum principle is applied to study a delayed
linear-quadratic optimal control problem with a non-convex control domain; an
optimal solution is obtained.
