Maximum principle for mean-field optimal control problems of delayed stochastic systems involving continuous and impulse controls

A Pontryagin's maximum principle for mean-field stochastic delay optimal control problems involving both continuous and impulse controls is considered in this paper. The solutions for a mean-field stochastic delay differential equation and a mean-field advanced backward stochastic differential equation are discussed, respectively. Then, a necessary maximum principle and a sufficient verification theorem by the duality method and by adding some convexity conditions are established, respectively.