• Corpus ID: 214743428

Superposition and mimicking theorems for conditional McKean-Vlasov equations

@article{Lacker2020SuperpositionAM,
  title={Superposition and mimicking theorems for conditional McKean-Vlasov equations},
  author={Daniel Lacker and Mykhaylo Shkolnikov and Jiacheng Zhang},
  journal={arXiv: Probability},
  year={2020}
}

• D. Lacker, Mykhaylo Shkolnikov, Jiacheng Zhang
• Published 31 March 2020
• Mathematics
• arXiv: Probability

We consider conditional McKean-Vlasov stochastic differential equations (SDEs), such 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 satisfy non-linear stochastic partial differential equations (SPDEs) of the second order, whereas the laws of the conditional time-marginals follow Fokker-Planck equations on the space of probability measures. We prove two superposition principles: The first establishes 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 Ito 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.