Killed path-dependent McKean-Vlasov SDEs for a probabilistic representation of non-conservative McKean PDEs

TL;DR

This paper develops a probabilistic framework for non-conservative, path-dependent PDEs using killed McKean-Vlasov SDEs, establishing existence, uniqueness, and regularity of solutions and their particle system counterparts.

Contribution

It introduces a novel approach linking killed McKean-Vlasov SDEs to non-conservative PDEs, proving well-posedness and regularity results.

Findings

Existence and uniqueness of strong solutions for the killed McKean-Vlasov SDEs.

Regularity properties of the sub-probability law's density.

Well-posedness of the associated particle system.

Abstract

A McKean-Vlasov stochastic differential equation subject to killing associated to a regularised non-conservative and path-dependent nonlinear parabolic partial differential equation is studied. The existence and pathwise uniqueness of a strong solution and the regularity properties of its sub-probability law are proved. The density of such a law may be seen as a weak solution of the considered PDE. The well-posedness of the associated particle system is also discussed.