Necessary Optimality Conditions for Weak Local Minima in Stochastic Control**The first author is partially supported by the Gaspard Monge Program for Optimisation(PGMO); the second author is partially supported by NSF of China under grants 11401404 and 11471231 and the fundamental research funds for the central universities under grants SWU114074 and XDJK2015C142; the third author is partially supported by NSF of China under grant 11231007, and by the Chang Jiang Scholars Program from Chinese Education Ministry.