CONVERGENCE ANALYSIS OF MACHINE LEARNING ALGORITHMS FOR THE NUMERICAL SOLUTION OF MEAN FIELD CONTROL AND GAMES: II-THE FINITE HORIZON CASE

• 成果类型: Article
• 署名作者: Carmona, Rene; Lauriere, Mathieu
• 署名单位: Princeton University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1715
• 发表日期: 2022
• 页码: 4065-4105
• 关键词: partial-differential-equations; mckean-vlasov; approximation

摘要

We propose two numerical methods for the optimal control of McKean- Vlasov dynamics in finite time horizon. Both methods are based on the intro-duction of a suitable loss function defined over the parameters of a neural net-work. This allows the use of machine learning tools, and efficient implemen-tations of stochastic gradient descent in order to perform the optimization. In the first method, the loss function stems directly from the optimal control problem. The second method tackles a generic forward-backward stochastic differential equation system (FBSDE) of McKean-Vlasov type, and relies on suitable reformulation as a mean field control problem. To provide a guaran-tee on how our numerical schemes approximate the solution of the original mean field control problem, we introduce a new optimization problem, di-rectly amenable to numerical computation, and for which we rigorously pro-vide an error rate. Several numerical examples are provided. Both methods can easily be applied to certain problems with common noise, which is not the case with the existing technology. Furthermore, although the first approach is designed for mean field control problems, the second is more general and can also be applied to the FBSDEs arising in the theory of mean field games.