## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 3229-3288 |

Number of pages | 60 |

Journal | Journal of the European Mathematical Society |

Volume | 25 |

Issue number | 8 |

DOIs | |

State | Published - 2023 |

## All Science Journal Classification (ASJC) codes

- General Mathematics
- Applied Mathematics

## Keywords

- Conditional McKean–Vlasov stochastic differential equations
- Fokker–Planck equations on the space of measures
- Markovian controls
- controlled McKean–Vlasov dynamics
- mean field games
- mimicking theorem
- particle systems with mean field interactions
- stochastic partial differential equations
- superposition theorem