On time delay optimal stochastic control problem of general McKean–Vlasov models with Poisson-jumps

This paper establishes a time delay necessary and sufficient stochastic maximum principle for a McKean–Vlasov model with randomness described by Brownian motions and Poisson jumps with noisy observation. The controlled delayed state process is governed by a general McKean–Vlasov nonlinear Itô stochastic differential equation with time delay (MVDSDE) driven by Poisson random jumps with correlate noisy observation. The coefficients of the controlled delay system depend on the state process as well as of its distribution and the control variable with time delay. Our main result is proved by applying convex perturbation method, approximate technique, Girsanov's lemma, and L-partial derivatives with respect to distribution. Finally, some examples with delay partially observed linear quadratic control problem of McKean–Vlasov type with Poisson jumps are studied, where we derive the explicit expression of the optimal control in feedback form.