Deep Learning for Mean Field Games and Mean Field Control with Applications to Finance

Abstract

Financial markets and more generally macro-economic models involve a large number of individuals interacting through variables such as prices resulting from the aggregate behavior of all the agents. Mean field games have been introduced to study Nash equilibria for such problems in the limit when the number of players is infinite. The theory has been extensively developed in the past decade, using both analytical and probabilistic tools, and a wide range of applications have been discovered, from economics to crowd motion. More recently the interaction with machine learning has attracted a growing interest. This aspect is particularly relevant to solve very large games with complex structures, in high dimension or with common sources of randomness. In this chapter, we review the literature on the interplay between mean field games and deep learning, with a focus on three families of methods. A special emphasis is given to financial applications.

Reinforcement Learning with Dynamic Convex Risk Measures

“…LeCun et al, 2015;Silver et al, 2016;Goodfellow et al, 2016), especially in financial mathematics (see e.g. Al-Aradi et al, 2018;Hu, 2019;Casgrain et al, 2019;Horvath et al, 2021;Campbell et al, 2021;Carmona and Laurière, 2021). The use of compositions of simple functions (usually referred to as propagation and activation functions) through several layers does a good job in modeling complicated functions.…”

Machine Learning architectures for price formation models

“…The interested reader is referred to e.g. [14] for a survey of deep learning methods applied to MFGs and mean-field control problems. However, the problem in (1.1) is quite distinct from the previous ones.…”

Deep Learning for Principal-Agent Mean Field Games

“…Similarly, [29] uses DL techniques to tackle MFC problems with delay. [14] provides an excellent overview of techniques in this area and their applications to finance.…”

“…The contributions of this work are threefold. Firstly, we further generalize the application of the deep BSDE method seen in [39,57,3,40,12,13,14] to MV-FBSDEs of the type that have dependence on the law of the adjoint process. This arises, for example, in settings where there is market clearing in addition to equilibrium (see, e.g., [55]).…”