Stochastic linear-quadratic optimal control problems with delay and Lévy processes

This paper is concerned with a linear-quadratic (LQ) optimal control problem for the stochastic system with delay and Lévy processes. To solve this problem, we study the solvability of stochastic Hamiltonian system which is a fully coupled anticipated forward-backward stochastic differential equation with delay and Lévy processes (AFBSDDEL). By virtue of solvability of AFBSDDEL, we give a representation of the unique optimal control with open-loop form. Moreover, the existence and uniqueness for the solution of AFBSDDEL in this paper can also be used to solve more general problem in practice such as game problem.