A.Bernou & Y. Liu: Path-dependent McKean-Vlasov equation: strong well-posedness, propagation of chaos and convergence of an interpolated Euler scheme

Abstract

We consider the path-dependent McKean-Vlasov equation, in which both the drift and the diffusion coefficients are allowed to depend on the whole trajectory of the process up to the current time t, and depend on the corresponding marginal distributions. We prove the strong well-posedness of the equation in the L^p setting, p greater or equal than 2, locally in time, as well as the propagation of chaos properties. Then, we introduce an interpolated Euler scheme, a key object to simulate numerically the process, and we prove the convergence, with an explicit rate, of this scheme towards the strong solution in the L^p norm. As applications we give results for two mean-field equations arising in biology and neuroscience.