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We study a mean-field version of Bank-El Karoui's representation theorem ofstochastic processes. Under different technical conditions, we establish someexistence and uniqueness results. As motivation and first applications, ourmean-field representation results provide a unified approach to study differentMean-Field Games (MFGs) in the setting with common noise and multiplepopulations, including the MFG of timing, the MFG with singular control, etc.As a crucial technical step, we provide a stability result on the classicalBank-El Karoui's representation theorem, which has its own interests and otherapplications, such as in deriving stability results of the optimizers (in thestrong sense) for a class of optimal stopping problems and singular controlproblems.

A mean-field version of Bank-El Karoui's representation of stochastic processes