A General Conditional McKean-Vlasov Stochastic Differential Equation

TL;DR

This paper introduces a new class of conditional McKean-Vlasov stochastic differential equations, establishing their well-posedness under certain conditions, and extends the theory to include state-dependent conditional laws.

Contribution

The paper develops a novel framework for conditional McKean-Vlasov SDEs with state-dependent laws, overcoming technical challenges related to non-compactness and nonlinearity.

Findings

Proved well-posedness of the new class of SDEs

Extended the theory to include state-dependent conditional laws

Addressed technical challenges with a direct approach

Abstract

In this paper we consider a class of {\it conditional McKean-Vlasov SDEs} (CMVSDE for short). Such an SDE can be considered as an extended version of McKean-Vlasov SDEs with common noises, as well as the general version of the so-called {\it conditional mean-field SDEs} (CMFSDE) studied previously by the authors [1, 14], but with some fundamental differences. In particular, due to the lack of compactness of the iterated conditional laws, the existing arguments of Schauder's fixed point theorem do not seem to apply in this situation, and the heavy nonlinearity on the conditional laws caused by change of probability measure adds more technical subtleties. Under some structure assumptions on the coefficients of the observation equation, we prove the well-posedness of solution in the weak sense along a more direct approach. Our result is the first that deals with McKean-Vlasov type SDEs involving state-dependent conditional laws.