Necessary optimality conditions of the second oder in a stochastic optimal control problem with delay argument

The optimal control problem of nonlinear stochastic systems which mathematical model is given by Ito stochastic differential equation with delay argument is considered. Assuming that the concerned region is open for the control by the first and the second variation (classical sense) of the quality functional we obtain the necessary optimality condition of the first and the second order. In the particular case we receive the stochastic analog of the Legendre—Clebsch condition and some constructively verified conclusions from the second order necessary condition. We investigate theLegendre–Clebsch conditions for the degeneration case and obtain the necessary conditions of optimality for a special control, in the classical sense.