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Pointwise second-order necessary conditions for stochastic optimal controls

In this talk, I will present some recent works on second-order necessary conditions for the stochastic optimal control problem, in the case that the control variable enters into both the drift and the diffusion terms. When the control set is convex, a pointwise second-order necessary condition for stochastic singular optimal controls in the classical sense is established ; while for the general case, we derive a pointwise second-order necessary condition for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle.

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