Differential inclusions with mean derivatives having extreme right-hand sides and optimal control

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First we prove the existence of solutions of some special stochastic differential inclusion with mean derivatives having lower semi-continuous right-hand sides that may not be convex. Then we show that among those solutions there is a solution that minimizes a certain cost criterion. After that this result is applied to investigation of controlled stochastic differential equations with feed back, whose right-hand sides take values in extreme sets of Hausdorff continuous set-valued vector field with bounded convex images. By reducing the equation to the inclusion of above-mentioned sort we prove that there exists a control that realizes the optimal solution of the inclusion as an optimal solution of the equation (an analogue of Filippov's theorem).