In this paper, we address the mixed stochastic \(H_{2}/H_{\infty }\) control problem for a mean-field jump-diffusion system, where the state equation is influenced by both a standard Brownian motion and a Poisson random martingale measure. The control input is treated as a leader, while the disturbance is considered a follower, employing the Stackelberg game approach. Our paper makes the following key contributions: (i) Initially, we treat the \(H_{\infty }\) problem as the follower problem. Utilizing the stochastic maximum principle, we obtain the optimal open-loop solution for the follower and derive an explicit feedback representation of the optimal control, associated with the follower problem. This feedback representation is obtained through coupled Riccati differential equations (CRDEs) using the Four-Step Scheme. (ii) Subsequently, we address the \(H_{2}\) problem as the leader problem. By applying the convex variational method, we establish the mean-field stochastic maximum principle. However, due to technical complexities related to the jump process, we consider two distinct situations. In both cases, we introduce new state variables and effectively employ the Four-Step Scheme to obtain the explicit feedback representation of the optimal control for the leader problem. (iii) We demonstrate that the open-loop Stackelberg equilibrium point can be characterized by a feedback representation, which incorporates both the state and its expected value. (iv) To enhance the practicality of our model, we present an example of a PA problem involving moral hazard, illustrating the corresponding optimal contract and optimal strategy. Therefore with our proposed methodologies and solutions, we contribute valuable insights into the mixed stochastic \(H_{2}/H_{\infty }\) control problem for mean-field jump-diffusion systems. And our findings offer significant implications for understanding and addressing such complex control problems in various real-world scenarios.

在本文中，我们解决了平均场跳跃扩散系统的混合随机\(H_{2}/H_{\infty }\)控制问题，其中状态方程同时受到标准布朗运动和泊松运动的影响随机鞅测度。采用 Stackelberg 博弈方法，控制输入被视为领导者，而干扰被视为跟随者。我们的论文做出了以下关键贡献：（i）最初，我们将\(H_{\infty }\)问题视为跟随者问题。利用随机极大值原理，我们获得了跟随器的最优开环解，并导出了与跟随器问题相关的最优控制的显式反馈表示。该反馈表示是通过使用四步方案的耦合 Riccati 微分方程 (CRDE) 获得的。(ii) 随后，我们将\(H_{2}\)问题视为领导者问题。应用凸变分法，建立了平均场随机极大值原理。然而，由于与跳跃过程相关的技术复杂性，我们考虑两种不同的情况。在这两种情况下，我们引入了新的状态变量，并有效地采用四步方案来获得领导者问题最优控制的显式反馈表示。(iii) 我们证明开环 Stackelberg 平衡点可以通过反馈表示来表征，其中包含状态及其期望值。(iv) 为了增强模型的实用性，我们提出了一个涉及道德风险的 PA 问题的例子，说明了相应的最优契约和最优策略。因此，通过我们提出的方法和解决方案，我们为平均场跳跃扩散系统的混合随机\(H_{2}/H_{\infty }\)控制问题提供了有价值的见解。我们的研究结果对于理解和解决各种现实场景中此类复杂的控制问题具有重要意义。