Superposition and mimicking theorems for conditional McKean–Vlasov equations
Daniel Lacker
Columbia University, New York, USAMykhaylo Shkolnikov
Princeton University, USAJiacheng Zhang
Princeton University, USA
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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.
Cite this article
Daniel Lacker, Mykhaylo Shkolnikov, Jiacheng Zhang, Superposition and mimicking theorems for conditional McKean–Vlasov equations. J. Eur. Math. Soc. 25 (2023), no. 8, pp. 3229–3288
DOI 10.4171/JEMS/1266