2020/03/31 by Daniel Lacker, Lacker, Daniel, Mykhaylo Shkolnikov +3 · 3 citations
Economics, Econometrics and Finance · Physics and Astronomy · #35Q84 #60H10 #60H15 #93E20 #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR) #Statistical Mechanics and Entropy #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2004.00099
openalex publication_date 2020/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.