Abstract
In this paper, we establish the second-order necessary conditions for optimal continuous-singular stochastic control, where the system is governed by nonlinear controlled Itô stochastic differential equation. The control process has two components, the first being absolutely continuous and the second of bounded variation, non-decreasing continuous on the right with left limits. Pointwise second-order maximum principle in terms of the martingale with respect to the time variable is proved. The control domain is assumed to be convex. In this paper, the continuous control variable enters into both the drift and the diffusion terms of the control systems. Our result is proved by using variational techniques under some convexity conditions.
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