Abstract
Mean field games are concerned with the limit of large-population stochastic differential games where the agents interact through their empirical distribution. In the classical setting, the number of players is large but fixed throughout the game. However, in various applications, such as population dynamics or economic growth, the number of players can vary across time and this may lead to different Nash equilibria. In order to account for this evolution, we introduce a branching mechanism in the population of agents and obtain a variant of the original mean field game problem. As a first step, we study a simple model using a PDE approach to illustrate the main differences with the classical setting. We prove existence of a solution and show that it provides an approximate Nash-equilibrium for large population games. We also present a numerical example for a linear–quadratic model. Then we study the problem in a general setting by a probabilistic approach. It is based upon the relaxed formulation of stochastic control problems which allows us to obtain a general existence result.
Funding Statement
The third author was supported in part by Hong Kong RGC General Research Fund (projects 14302921).
Acknowledgments
The second author thanks the support of Finance for Energy Market Research Initiative.
Citation
Julien Claisse. Zhenjie Ren. Xiaolu Tan. "Mean field games with branching." Ann. Appl. Probab. 33 (2) 1034 - 1075, April 2023. https://doi.org/10.1214/22-AAP1835
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