论文标题
平均田间随机控制在均匀期望下
Mean field stochastic control under sublinear expectation
论文作者
论文摘要
我们的工作致力于研究Pontryagin在Peng的$ G $期望下的均值最佳控制问题的随机最大原则。受控状态流程的动态由由$ g $ -Brownian运动驱动的随机微分方程给出，其系数不仅取决于控制，也取决于受控状态流程，而且还取决于$ G $ prection的法律。同样，相关的成本功能为平均场类型。在凸控制状态空间的假设下，我们研究了随机最大原理，这为控制过程提供了必要的最佳条件。在对哈密顿的额外凸度假设下，这表明这种必要的条件也足够。我们在工作中必须克服的主要困难在于分化了$ g $的参数化随机变量。特别是精致，事实证明，在成本功能的运行成本中，可以通过$ g $的函数来处理。为此，我们必须研究针对设定值函数的可测量选择定理，其值是$ g $ - 期望的代表概率度量集的子集的子集。
Our work is devoted to the study of Pontryagin's stochastic maximum principle for a mean-field optimal control problem under Peng's $G$-expectation. The dynamics of the controlled state process is given by a stochastic differential equation driven by a $G$-Brownian motion, whose coefficients depend not only on the control, the controlled state process but also on its law under the $G$-expectation. Also the associated cost functional is of mean-field type. Under the assumption of a convex control state space we study the stochastic maximum principle, which gives a necessary optimality condition for control processes. Under additional convexity assumptions on the Hamiltonian it is shown that this necessary condition is also a sufficient one. The main difficulty which we have to overcome in our work consists in the differentiation of the $G$-expectation of parameterized random variables. As particularly delicate it turns out to handle with the $G$-expectation of a function of the controlled state process inside the running cost of the cost function. For this we have to study a measurable selection theorem for set-valued functions whose values are subsets of the representing set of probability measures for the $G$-expectation.
