A general conditional McKean–Vlasov stochastic differential equation

Abstract

In this paper we consider a class of 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 conditional mean-field SDEs (CMFSDE) studied previously by the authors (Ann. Appl. Probab. 27 (2017) 3201–3245; SIAM J. Control Optim. 56 (2018) 1154–1180), 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 structural assumptions on the coefficients of the observation equation, we prove the well-posedness of the solutions in a weak sense along a more direct approach. Our result is the first that deals with McKean–Vlasov type SDEs involving state-dependent conditional laws.