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The main purpose of this paper is to establish the first and second ordernecessary optimality conditions for stochastic optimal controls using theclassical variational analysis approach. The control system is governed by astochastic differential equation, in which both drift and diffusion terms maycontain the control variable and the set of controls is allowed to benonconvex. Only one adjoint equation is introduced to derive the first ordernecessary condition; while only two adjoint equations are needed to state thesecond order necessary conditions for stochastic optimal controls.

First and second order necessary conditions for stochastic optimal controls