On optimal singular control problem for general Mckean-Vlasov differential equations: Necessary and sufficient optimality conditions

In this paper, we derive the necessary and sufficient conditions for optimal singular control for systems governed by general controlled McKean-Vlasov differential equations, in which the coefficients depend on the state of the solution process as well as of its law and control. The control domain is assumed to be convex. The control variable has 2 components, ie, the first being absolutely continuous and the second being singular. The proof of our result is based on the derivative of the solution process with respect to the probability law and a corresponding Ito formula. Finally, an example is given to illustrate the theoretical results.