Abstract
This paper derives necessary and sufficient conditions of optimality in the form of a stochastic maximum principle for relaxed and strict optimal control problems with jumps. These problems are governed by multi-dimensional forward-backward doubly stochastic differential equations (FBDSDEs) with Poisson jumps and has firstly relaxed controls, which are measure-valued processes, and secondly, as an application, the authors allow them to have strict controls. The FBDSDEs with jumps are fully-coupled, the forward and backward equations work in different Euclidean spaces in general, the backward equation is Markovian, and the control problems are considered under full information or partial information in terms of sigma-algebras that provide such information. The formulation of these equations as well as performance functionals are given in abstract forms to allow the possibility to cover most of the applications available in the literature. Moreover, coefficients of such equations are allowed to depend on control variables.