Maximum principle for anticipated recursive stochastic optimal control problem with delay and Lévy processes

In this paper, we study the stochastic maximum principle for optimal control problem of anticipated forward-backward system with delay and Lévy processes as the random disturbance. This control system can be described by the anticipated forward-backward stochastic differential equations with delay and Lévy processes (AFBSDEDLs), we first obtain the existence and uniqueness theorem of adapted solutions for AFBSDEDLs; combining the AFBSDEDLs’ preliminary result with certain classical convex variational techniques, the corresponding maximum principle is proved.