Mean Field Games: Stochastic Control and Applications to Trading
Summary
The document introduces differential games and mean field games as frameworks that combine game theory with stochastic calculus. It describes two research origins: the mean field game work of Lasry and Lions, and large-population stochastic dynamic games associated with the Nash certainty-equivalence principle. It points to applications including high-frequency price formation, trade crowding, and optimal execution.
Its intuitive outline treats an agent’s stochastic control problem as depending on the population distribution. Solving the control problem yields optimal actions; those actions then move the distribution forward through a PDE. Repeating the solve-and-update steps may converge to a fixed point representing a consistent population strategy. The account is only schematic: it gives no equations, convergence conditions, or evidence that iteration succeeds in a particular market setting, and the listed papers serve as suggested further reading.
Key ideas
- Mean field games model strategic interactions among large populations of agents.
- An individual’s stochastic control problem can depend on the distribution of other agents.
- Optimal controls move the population distribution forward through a PDE.
- Iterating control solutions and distribution updates may reach a consistent fixed point.
- Finance applications include price formation, trading crowding, and optimal execution.
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# Game theory and stochastic calculus # Game theory and stochastic calculus Does anybody know any details of game theory literature combined with stochastic calculus in finance? If yes, please recommend some papers of any authors who are doing exceptional work on the filed. Thank you in advance! ## Answer by lehalle (score 2) https://quant.stackexchange.com/a/68856 The field you have in mind is covered with differential game theory, and it game birth to Mean Field Games (MFG), the book posted in a comment is certainly the reference: Probabilistic Theory of Mean Field Games with Applications volume 1 and 2 by Carmona and Delarue. MFG started with two independent trends of research: - Mean field games by Lasry and Lions - Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle by Huang, Malhamé, and Caines. It is now applied to finance, the first paper doing it has been Efficiency of the Price Formation Process in Presence of High Frequency Participants: a Mean Field Game analysis by Lachapelle, Lasry, L and Pierre-Louis Lions. Later on, different authors applied it to trading flows, see - Mean Field Game of Controls and An Application To Trade Crowding, by Cardaliaguet and L, - Mean-Field Game Strategies for Optimal Execution by Huang, Jaimungal and Nourian. A simple way to understand MFG. Say you have a stochastic control problem on each agent of a game, and that the value function of all agents is a function of the "positions" of all other agents. For instance each agent could desire to be away from this other, but to have an incentive that the standard deviation of the distribution of all agents is not too large. (step 0) You can express this problem as a standard stochastic problem involving the density $m$ of all the agents. You can solve it as a function of $m_0$ that is an arbitrary density to start with. Now you have to consider that each agent is implementing its optimal strategies: - whereas the solution of the stochastic control problem is backward - now the controls of the agents act as a "push forward" of their positions; it is a forward PDE. (step 1) It is possible to "run" this PDE on the distribution of the agents: it gives you a new position $m_1$ of these agents. And you iterate steps (0-1), ultimately is may converge to a fixed point (as usual in game theory) that will indeed be the optimal positions of the agents.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.