Necessary conditions for partially observed optimal control of general McKean-Vlasov stochastic differential equations with jumps

In this paper, we establish necessary conditions of optimality for partially observed optimal control problems of Mckean-Vlasov type. The system is described by a controlled stochastic differential equation governed by Poisson random measure and an independent Brownian motion. The coefficients of the McKean-Vlasov system depend on the state of the solution process as well as of its probability law and the control variable. The proof of our result is based on Girsanov's theorem, variational equations and derivatives with respect to probability measure under convexity assumption. At the end of this paper, we apply our stochastic maximum principle to study partially observed linear quadratic control problem of McKean-Vlasov type with jumps and derive the explicit expression of the optimal control.