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 σ-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.

本文以随机最大原理的形式导出了具有松弛的严格控制问题的最优性的充要条件。这些问题由具有泊松跳跃的多维向前-向后双重随机微分方程（FBDSDE）控制，首先具有宽松的控件，这些控件是度量值过程，其次，作为应用，作者允许他们使用严格的控件。具有跳跃的FBDSDE是完全耦合的，前向和后向方程通常在不同的欧几里得空间中工作，后向方程是马尔可夫方程，并且根据提供这些信息的σ代数，在完全信息或部分信息下考虑了控制问题。 。这些方程式以及性能函数的表述以抽象形式给出，以允许涵盖文献中可用的大多数应用程序。此外，这种等式的系数被允许取决于控制变量。