A Novel Approach to Peng's Maximum Principle for McKean-Vlasov Stochastic Differential Equations

TL;DR

This paper introduces a new proof of Peng's maximum principle for McKean-Vlasov SDEs by adding a third adjoint equation, facilitating extensions to more complex stochastic systems.

Contribution

It proposes a novel approach using a third adjoint equation, extending Peng's maximum principle to McKean-Vlasov SDEs with potential for broader applications.

Findings

Introduces a third adjoint equation for McKean-Vlasov SDEs.

Provides a conceptually consistent proof of the maximum principle.

Facilitates extensions to common noise and infinite-dimensional settings.

Abstract

We present a novel approach to the proof of Peng's maximum principle for McKean-Vlasov stochastic differential equations (SDE). The main step is the introduction of a third adjoint equation, a conditional McKean-Vlasov backward SDE, to accommodate the dualization of quadratic terms containing two independent copies of the first-order variational process. This is an intrinsic extension of the maximum principle from Peng for standard SDE and gives a conceptually consistent proof. Our approach will be useful in further extensions to the common noise setting and the infinite dimensional setting.