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 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 Malliavin calculus, we establish the desired necessary condition for stochastic singular optimal controls in the classical sense. 
Haisen Zhang and Xu Zhang
math.OC