Stochastic Maximum Principle for Mean‐Field Delayed Backward Doubly Stochastic Differential Equations

In this paper, we focus on a kind of mean‐field doubly stochastic optimal control problems described by mean‐field delayed backward doubly stochastic differential equations. To begin with, we obtain the existence and uniqueness result of the solution for mean‐field delayed backward doubly stochastic differential equations (MFDBDSDEs, for short) and mean‐field anticipated backward doubly stochastic differential equations (MFABDSDEs, for short) by the contraction mapping principle, respectively. Then, under the convexity condition, we deduce the stochastic maximum principle of the mean‐field delayed backward doubly stochastic control systems by utilizing the classical variational theory. After that, we give the sufficient conditions for the stochastic control systems by using the duality relation between the state equations and the corresponding adjoint equations. Finally, we give the application to the mean‐field delayed doubly stochastic linear quadratic optimal control problems.