AI 中文总结
本文建立次扩散驱动的平均场随机微分方程及倒向方程解的存在唯一性，并利用凸变分方法推导随机最大值原理，通过线性二次控制示例证明最优控制的存在唯一性。
AI 中文摘要
本文分别建立了由具有随机系数的反常次扩散$\{B_{L_t}; t\geq 0\}$驱动的平均场随机微分方程（简称MF-SDEs）和倒向随机微分方程（MF-BSDE）解的存在唯一性。这里$B$是${\mathbb R}^d$上的布朗运动，$L$是漂移$\kappa >0$且与$B$独立的子ordinator $S$的逆。我们进一步利用凸变分方法研究了由MF-SDEs建模的随机系统的控制问题的随机最大值原理（SMPs）。本文最后一节给出了一个线性二次控制示例，其中利用本文建立的SMP和充分SMP明确证明了该示例存在唯一的最优随机控制。
英文摘要
In this paper, we establish the existence and uniqueness of solutions for mean-field stochastic differential equations (MF-SDEs in short) and backward stochastic differential equations (MF-BSDE) driven by anomalous sub-diffusions $\{B_{L_t}; t\geq 0\}$ with random coefficients, respectively. Here $B$ is a Brownian motion on ${\mathbb R}^d$ and $L$ is the inverse of a subordinator $S$ with drift $κ>0$ that is independent of $B$. We further study the stochastic maximum principles (SMPs) for control problems of the stochastic systems modelled by the MF-SDEs using a convex variational method. A linear quadratic control example is given in the last section of this paper, for which both the SMP and the sufficient SMP established in this paper are utilized to show explicitly that it admits a unique stochastic optimal control.
