A linear stochastic control system with scalar controls constrained to remain in prescribed intervals is considered. The objective is to minimize a finite horizon cost with nonlinear, noisy running cost and a nonlinear deterministic terminal cost on a finite horizon. This is mapped to a constrained optimization problem analyzed using methods of set-valued variational analysis. The notions of Mordukovich normal cone and Mordukovich coderivative are introduced and used for formulating optimality conditions in terms of a `parametric variational inequality'. An explicit formula for the Mordukovich coderivative is derived and illustrated with an example. The notion of `local metric regularity in the sense of Robinson' for the solution map is defined in terms of the above and verified for the control problem under consideration.