This manuscript investigates a broad class of stochastic differential equation steered by Poisson jumps and impulses in a Hilbert space. We use successive approximation method to show the existence of a solution. The continuous dependence of the solutions to the initial data is also studied. Finally, the existence of optimal state-control pair for the associated Lagrangian problem is discussed. An example is also presented to validate the theoretical results.

中文翻译：

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具有泊松跳跃的无限延迟随机脉冲微分系统

这份手稿研究了希尔伯特空间中由泊松跳跃和脉冲控制的一类广泛的随机微分方程。我们使用逐次逼近法来证明解的存在性。还研究了解对初始数据的连续依赖性。最后，讨论了相关拉格朗日问题的最优状态-控制对的存在性。还提供了一个例子来验证理论结果。