Necessary conditions for quasi-singular controls in the stochastic optimal control problem with delayed argument

The optimal control problem is considered, the mathematical models of which are defined by non-linear stochastic Ito differential equations with a delay argument and diffuse components that allow one to take into account random disturbances of a continuous nature acting on the system.
A linearized necessary optimality condition is obtained under the assumption that the domain admissible control is convex. The quasi-singular case is investigated. The general necessary optimality conditions for quasi-singular controls are described. Partial cases are considered.