Let $(E,\mathcal{E})$ be an arbitrary measurable space. The paper first focuses on studying the partial derivative of a function $f:{\mathcal{P}}_{2,0}({\mathbb{R}}^d\times E)\to \mathbb{R}$ defined on the space of probability measures $\mathrm{\mu }$ over $({\mathbb{R}}^d\times E,\mathcal{B}\left({\mathbb{R}}^d\right)\otimes \mathcal{E})$ whose first marginal ${\mu }_1≔\mathrm{\mu }(\cdot \times E)$ has a finite second order moment. This partial derivative is taken with respect to $q(dx,z)$, where $\mathrm{\mu }$ has the disintegration $\mathrm{\mu }\left(dxdz\right)=q(dx,z){\mathrm{\mu }}_2\left(dz\right)$ with respect to its second marginal ${\mu }_2\left(\cdot \right)=\mathrm{\mu }({\mathbb{R}}^d\times \cdot )$. Simplifying the language, we will speak of the derivative with respect to the law $\mathrm{\mu }$ conditioned to its second marginal. Our results extend those of the derivative of a function $g:{\mathcal{P}}_2\left({\mathbb{R}}^d\right)\to \mathbb{R}$ over the space of probability measures with finite second order moment by P.L. Lions (see Lions (2013)) but cover also as a particular case recent approaches considering $E={\mathbb{R}}^k$ and supposing the differentiability of $f$ over ${\mathcal{P}}_2({\mathbb{R}}^d\times {\mathbb{R}}^k)$, in order to use the derivative ${\partial }_{\mathrm{\mu }}f$ to define the partial derivative ${\left({\partial }_{\mathrm{\mu }}f\right)}_1$. The second part of the paper focuses on investigating a stochastic maximum principle, where the controlled state process is driven by a general mean-field stochastic differential equation with partial information. The control set is just supposed to be a measurable space, and the coefficients of the controlled system, i.e., those of the dynamics as well as of the cost functional, depend on the controlled state process $X$, the control $v$, a partial information on $X$, as well as on the joint law of $(X,v)$. Through considering a new second-order variational equation and the corresponding second-order adjoint equation, and a totally new method to prove the estimate for the solution of the first-order variational equation, the optimal principle is proved through spike variation of an optimal control and with the help of the tailor-made form of second-order expansion. We emphasize that in our assumptions we do not need any regularity of the coefficients neither in the control variable nor with respect to the law of the control process.

让 $（Ë，\mathcal{Ë}）$是一个任意可测量的空间。本文首先着重研究函数的偏导数$F：{\mathcal{P}}_{2，0}（{\mathbb{[R}}^d\times Ë）\to \mathbb{[R}$ 在概率测度空间上定义 $\mathrm{\mu }$ 过度 $（{\mathbb{[R}}^d\times Ë，\mathcal{乙}（{\mathbb{[R}}^d）\otimes \mathcal{Ë}）$ 他的第一边际 ${\mu }_{\mathrm{1个}}≔\mathrm{\mu }（\cdot \times Ë）$具有有限的二阶矩 该偏导数是相对于$q（dX，ž）$，在哪里 $\mathrm{\mu }$ 有瓦解 $\mathrm{\mu }（dXdž）=q（dX，ž）{\mathrm{\mu }}_2（dž）$ 关于其第二边缘 ${\mu }_2（\cdot ）=\mathrm{\mu }（{\mathbb{[R}}^d\times \cdot ）$。简化语言，我们将谈谈关于法律的派生词$\mathrm{\mu }$限制在第二边缘。我们的结果扩展了函数导数的结果$G：{\mathcal{P}}_2（{\mathbb{[R}}^d）\to \mathbb{[R}$ PL Lions（参见Lions（2013））在具有有限二阶矩的概率测度空间上进行了研究，但作为一个特殊案例也涵盖了最近的方法 $Ë={\mathbb{[R}}^{ķ}$ 并假设 $F$ 过度 ${\mathcal{P}}_2（{\mathbb{[R}}^d\times {\mathbb{[R}}^{ķ}）$，以便使用导数 ${\partial }_{\mathrm{\mu }}F$ 定义偏导数 ${（{\partial }_{\mathrm{\mu }}F）}_{\mathrm{1个}}$。本文的第二部分着重研究随机最大原理，其中控制状态过程由具有部分信息的一般平均场随机微分方程驱动。控制集只是一个可测量的空间，受控系统的系数（即动力学系数和成本函数的系数）取决于受控状态过程$X$，控制 $v$，有关的部分信息 $X$以及关于 $（X，v）$。通过考虑一个新的二阶变分方程和相应的二阶伴随方程，以及一种全新的方法来证明一阶变分方程解的估计，通过最优控制的尖峰变化来证明最优原理。并借助量身定制的二阶扩展形式。我们强调，在我们的假设中，我们既不需要控制变量中的系数的任何规律性，也不需要控制过程的规律方面的系数。