论文标题
McKean-Vlasov最佳控制：不同公式之间的极限理论和等价
McKean-Vlasov optimal control: limit theory and equivalence between different formulations
论文作者
论文摘要
我们研究了McKean-Vlasov与共同噪声的最佳控制问题，以建立相应的极限理论以及不同配方之间的等效性，包括强，弱和放松的配方。与强大的配方相反，该问题是在配备了两个布朗过滤的固定概率空间上提出的，弱公式是通过考虑更一般的概率空间，其中有两个过滤，从而使$（h）$ - 假设类型的条件从过滤的扩大理论中获得。当不受控制的常见噪声时，我们的放松配方是通过考虑合适的受控玛格尔问题获得的。至于经典的最佳控制问题，我们证明所有放松控制的集合是在规范空间上的概率度量时，所有强大控制的集合都是封闭的。因此，在奖励功能的其他轻度规律条件下，我们获得了控制问题的不同公式的等效性。这也是证明McKean-Vlasov控制问题的极限理论的关键技术步骤，也就是说，证明它是由常见噪声带来的大种群控制问题的极限。
We study a McKean-Vlasov optimal control problem with common noise, in order to establish the corresponding limit theory, as well as the equivalence between different formulations, including the strong, weak and relaxed formulation. In contrast to the strong formulation, where the problem is formulated on a fixed probability space equipped with two Brownian filtrations, the weak formulation is obtained by considering a more general probability space with two filtrations satisfying an $(H)$-hypothesis type condition from the theory of enlargement of filtrations. When the common noise is uncontrolled, our relaxed formulation is obtained by considering a suitable controlled martingale problem. As for classical optimal control problems, we prove that the set of all relaxed controls is the closure of the set of all strong controls, when considered as probability measures on the canonical space. Consequently, we obtain the equivalence of the different formulations of the control problem, under additional mild regularity conditions on the reward functions. This is also a crucial technical step to prove the limit theory of the McKean-Vlasov control problem, that is to say proving that it consists in the limit of a large population control problem with common noise.
