Abstract
We formulate and analyze a mathematical framework for continuous-time mean field games with finitely many states and common noise. The key insight is that we can circumvent the master equation and reduce the mean field equilibrium to a system of forward-backward systems of (random) ordinary differential equations by conditioning on common noise events. In the absence of common noise, our setup reduces to that of Gomes, Mohr and Souza (2013) and Cecchin and Fischer (2020).
Highlights
Mean field games constitute a class of dynamic, multi-player stochastic differential games with identical agents
In solving an individual agent’s optimization problem, the feedback effect of his own actions on the aggregate outcome can be discarded, breaking the notorious vicious circle (“the optimal strategy depends on the aggregate outcome, which depends on the strategy, which depends ...”)
We provide a rigorous formulation of the underlying stochastic dynamics, and we establish a verification theorem for the optimal strategy and an aggregation theorem to determine the resulting aggregate distribution
Summary
Since the seminal contributions of Lasry and Lions [44] and Huang, Malhamé and Caines [39], mean field games have become an active field of mathematical research with a wide range of applications, including economics [13,16,27,33,41,50], sociology. In the presence of common noise, i.e. sources of risk that affect all agents and do not average out in the mean field limit, the mathematical analysis becomes even more involved as the dynamics of the aggregate distribution become stochastic, leading to conditional McKean-Vlasov dynamics; see, e.g., [1,12,21,51]. We set up a mathematical framework for finite-state mean field games with common noise.. We provide a rigorous formulation of the underlying stochastic dynamics, and we establish a verification theorem for the optimal strategy and an aggregation theorem to determine the resulting aggregate distribution. This leads to a characterization of the mean field equilibrium in terms of a system of (random) forward-backward differential equations. We provide the proofs of Theorems 13 and 16
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