Superposition and mimicking theorems for conditional McKean–Vlasov equations

We consider conditional McKean–Vlasov stochastic differential equations (SDEs), as the ones arising in the large-system limit of mean field games and particle systems with mean field interactions when common noise is present. The conditional time-marginals of the solutions to these SDEs are governed by non-linear stochastic partial differential equations (SPDEs) of the second order, whereas their laws satisfy Fokker–Planck equations on the space of probability measures. Our paper establishes two superposition principles: The first asserts that any solution of the SPDE can be lifted to a solution of the conditional McKean–Vlasov SDE, and the second guarantees that any solution of the Fokker–Planck equation on the space of probability measures can be lifted to a solution of the SPDE. We use these results to obtain a mimicking theorem which shows that the conditional time-marginals of an Itô process can be emulated by those of a solution to a conditional McKean–Vlasov SDE with Markovian coefficients. This yields, in particular, a tool for converting open-loop controls into Markovian ones in the context of controlled McKean–Vlasov dynamics.