Abstract The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of fractional neutral stochastic differential equations (NSDEs) driven by fractional Brownian motion and Poisson jumps in Hilbert spaces. First, we establish a new set of sufficient conditions for the existence of mild solutions of the aforementioned fractional systems by using the successive approximation approach. The results are formulated and proved by using the fractional calculus, solution operator, and stochastic analysis techniques. The existence of optimal control pairs of system governed by fractional NSDEs driven by fractional Brownian motion and Poisson jumps is also been presented. An example is provided to illustrate the theory.

中文翻译：

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带有 Caputo 分数阶导数的分数中性随机微分方程：分数布朗运动、泊松跳跃和最优控制

摘要 本文的目的是研究希尔伯特空间中由分数布朗运动和泊松跳跃驱动的一类分数中性随机微分方程（NSDE）的温和解和最优控制的存在性。首先，我们通过使用逐次逼近方法为上述分数系统的温和解的存在建立了一组新的充分条件。通过使用分数阶微积分、解算子和随机分析技术来制定和证明结果。还介绍了由分数布朗运动和泊松跳跃驱动的分数 NSDE 控制的系统的最优控制对的存在。提供了一个例子来说明该理论。