Abstract: I’m going to present two relevant topics in Mckean-Vlasov equation and mimicking theorem. The first one arises when one tries to invert the Markovian projection developed by Gyöngy (1986), typically to produce an Itô process with the fixed-time marginal distributions of a given one-dimensional diffusion but richer dynamical features. These SDEs enjoy frequent application in the calibration of local stochastic volatility models in finance, despite the very limited theoretical understanding. We prove the strong existence of stationary solutions for these SDEs, as well as their strong uniqueness in an important special case.

The second one arises 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 and the result can obtain the 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.