论文标题
无限维度中依赖路径依赖的麦基恩 - 弗拉索夫SDE的最佳控制
Optimal control of path-dependent McKean-Vlasov SDEs in infinite dimension
论文作者
论文摘要
我们研究了由由随机PDES驱动的非马尔克维亚平均模型动机的希尔伯特空间中有价值的路径依赖性麦基vlasov方程的最佳控制。我们首先建立了状态方程的良好性，然后在这样的一般框架中证明了动态编程原理（DPP）。严格获得了值函数V的关键定律不变特性，这意味着V可以将V视为Hilbert空间中价值连续函数集的Wasserstein概率测量空间上的函数。然后，我们定义了一个路径度量衍生物的概念，该概念扩展了由于狮子的[41]而扩展了瓦斯恒星的衍生物，并在dupire [24]和wu and Zhang [51]中证明了相关的功能IT {佛。主钟者方程是通过粘度解决方案的合适概念从DPP得出的。当不依赖控制法律时，我们在特殊情况下特别是在特殊情况下提供了不同的表述和简化。
We study the optimal control of path-dependent McKean-Vlasov equations valued in Hilbert spaces motivated by non Markovian mean-field models driven by stochastic PDEs. We first establish the well-posedness of the state equation, and then we prove the dynamic programming principle (DPP) in such a general framework. The crucial law invariance property of the value function V is rigorously obtained, which means that V can be viewed as a function on the Wasserstein space of probability measures on the set of continuous functions valued in Hilbert space. We then define a notion of pathwise measure derivative, which extends the Wasserstein derivative due to Lions [41], and prove a related functional It{ô} formula in the spirit of Dupire [24] and Wu and Zhang [51]. The Master Bellman equation is derived from the DPP by means of a suitable notion of viscosity solution. We provide different formulations and simplifications of such a Bellman equation notably in the special case when there is no dependence on the law of the control.
