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This paper is the first part of our series work to establish pointwisesecond-order necessary conditions for stochastic optimal controls. In thispart, both drift and diffusion terms may contain the control variable but thecontrol region is assumed to be convex. Under some assumptions in terms ofMalliavin calculus, we establish the desired necessary condition for stochasticsingular optimal controls in the classical sense.

Pointwise second-order necessary conditions for stochastic optimal controls, Part I: The case of convex control constraint