A general conditional McKean–Vlasov stochastic differential equation

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