Pointwise Second-order Necessary Conditions for Stochastic Optimal Controls, Part I: The Case of Convex Control Constraint

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

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