This paper investigates a stochastic optimal control problem with delay and of mean-field type, where the controlled state process is governed by a mean-field jump-diffusion stochastic delay differential equation. Two sufficient maximum principles and one necessary maximum principle are established for the underlying systems. As an application, a bicriteria mean-variance portfolio selection problem with delay is studied. Under certain conditions, explicit expressions are provided for the efficient portfolio and the efficient frontier, which are as elegant as those in the classical mean-variance problem without delays.

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A general maximum principle is proved for optimal controls of abstract semilinear stochastic evolution equations. The control variable and linear unbounded operators act in both drift and diffusion terms, and the control set need not be convex. Abstract. A general maximum principle is proved for optimal controls of abstract semilinear stochastic evolution equations. The control variable, as well as linear unbounded operators, acts in both drift and diffusion terms, and the control set need not be convex.

a b s t r a c tThis paper is concerned with an optimal control problem under mean-field jump-diffusion systems with delay. Firstly, some existence and uniqueness results are proved for a jumpdiffusion mean-field stochastic delay differential equation and a jump-diffusion meanfield advanced backward stochastic differential equation. Then necessary and sufficient maximum principles for control systems of mean-field type and with delay are established under certain conditions. A mean-field, delayed, linear-quadratic control problem is finally discussed using the obtained maximum principles.

In this paper, we study a linear-quadratic optimal control problem for mean-field stochastic differential equations driven by a Poisson random martingale measure and a one-dimensional Brownian motion. Firstly, the existence and uniqueness of the optimal control is obtained by the classic convex variation principle. Secondly, by the duality method, the optimality system, also called the stochastic Hamilton system which turns out to be a linear fully coupled mean-field forward-backward stochastic differential equation with jumps, is derived to characterize the optimal control. Thirdly, applying a decoupling technique, we establish the connection between two Riccati equations and the stochastic Hamilton system and then prove the optimal control has a state feedback representation.