Well-Posedness for Path-Distribution Dependent Stochastic Differential Equations with Singular Drifts

Abstract

Well-posedness is derived for singular path-distribution dependent stochastic differential equations (SDEs) with non-degenerate noise, where the drift is allowed to be singular in the current state, but maintains local Lipschitz continuity in the historical path, and the coefficients are Lipschitz continuous with respect to a weighted variation distance in the distribution variable. Notably, this result is new even for classical path-dependent SDEs where the coefficients are distribution independent. Moreover, by strengthening the local Lipschitz continuity to Lipschitz continuity and replacing the weighted variation distance with the Wasserstein distance, we also obtain well-posedness.