Extended Mean Field Control Problems: Stochastic Maximum Principle and Transport Perspective
Extended Mean Field Control Problems: Stochastic Maximum Principle and Transport Perspective
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DOI：
10.1137/18m1196479
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发表时间：
2018-02
期刊：
影响因子：
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通讯作者：
Beatrice Acciaio;Julio D. Backhoff Veraguas;R. Carmona
中科院分区：
文献类型：
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作者：
Beatrice Acciaio;Julio D. Backhoff Veraguas;R. Carmona
我们研究平均场随机控制问题，其中成本函数和状态动态取决于受控状态和控制过程的联合分布。我们以必要和充分的形式证明了庞特里亚金随机极大值原理的合适版本，将已知条件扩展到了这个一般框架。此外，我们建议采用变分方法来研究这些控制问题的弱表述。我们展示了这种弱公式与路径空间上的最优传输之间的自然联系，这激发了一种新颖的离散化方案。
We study Mean Field stochastic control problems where the cost function and the state dynamics depend upon the joint distribution of the controlled state and the control process. We prove suitable versions of the Pontryagin stochastic maximum principle, both in necessary and in sufficient form, which extend the known conditions to this general framework. Furthermore, we suggest a variational approach to study a weak formulation of these control problems. We show a natural connection between this weak formulation and optimal transport on path space, which inspires a novel discretization scheme.