April 2023 Mean field games with branching
Julien Claisse, Zhenjie Ren, Xiaolu Tan
Author Affiliations +
Ann. Appl. Probab. 33(2): 1034-1075 (April 2023). DOI: 10.1214/22-AAP1835


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).


The second author thanks the support of Finance for Energy Market Research Initiative.


Download 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


Received: 1 April 2021; Revised: 1 April 2022; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 1519.91033
MathSciNet: MR4564420
Digital Object Identifier: 10.1214/22-AAP1835

Primary: 60J80
Secondary: 91A13 , 93E20

Keywords: branching diffusion process , Mean field games , relaxed control

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 2 • April 2023
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