Integration by Parts Formulas of Mckean-Vlasov SDEs with Jumps and Some Applications

TL;DR

This paper develops integration by parts formulas for McKean-Vlasov SDEs with jumps, enabling derivative estimates of densities and solutions to related PDEs with irregular conditions.

Contribution

It introduces new integration by parts formulas for McKean-Vlasov SDEs with jumps, accommodating derivatives in both real and measure variables.

Findings

Derived estimates for density derivatives of McKean-Vlasov SDEs.

Proved existence and uniqueness of classical solutions to associated PDEs with irregular terminal conditions.

Abstract

In this article, we establish integration by parts formulas for the solutions of McKean-Vlasov stochastic differential equations with jumps under elliptic coefficients. The derived formulas accommodate both derivatives with respect to real-valued variables and measure-valued variables, interpreted through the Lions' derivative. As applications, we obtain estimates for the derivatives of the density functions of the McKean-Vlasov SDEs, and relying on the integration by parts formulas, we subsequently prove the existence and uniqueness of classical solutions to the associated PDEs with irregular terminal conditions.