In this paper, we consider a problem of optimal control of an infinite horizon mean-field backward stochastic differential equation with delay and noisy memory under partial information. We derive necessary and sufficient maximum principles using Malliavin calculus technique for such a system. A class of mean-field time-advanced stochastic differential equations is introduced as the adjoint process which involves partial derivatives of the Hamiltonian functions and their Malliavin derivatives. To illustrate our theoretical results, we give an example for a linear-quadratic mean-field backward delay stochastic system with noisy memory on infinite horizon to obtain the optimal control. Also, we apply our results to pension fund problems with delay and noisy memory which are arising from the financial market.

在本文中，我们考虑了在部分信息下具有延迟和噪声记忆的无限水平平均场反向随机微分方程的最优控制问题。我们使用 Malliavin 微积分技术为这样的系统推导出必要和充分的最大原则。引入了一类平均场时间超前随机微分方程作为伴随过程，它涉及哈密顿函数的偏导数及其Malliavin导数。为了说明我们的理论结果，我们给出了一个线性二次平均场后向延迟随机系统的例子，该系统在无限范围内具有噪声记忆以获得最佳控制。此外，我们将我们的结果应用于由金融市场引起的具有延迟和嘈杂记忆的养老基金问题。