June 2023 A general conditional McKean–Vlasov stochastic differential equation
Rainer Buckdahn, Juan Li, Jin Ma
Author Affiliations +
Ann. Appl. Probab. 33(3): 2004-2023 (June 2023). DOI: 10.1214/22-AAP1858


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.

Funding Statement

Rainer Buckdahn is supported by the ANR (Agence Nationale de la Recherche), France project ANR-16-CE40-0015-01. Juan Li is supported by the NSF of P.R. China (NOs. 12031009, 11871037), National Key R and D Program of China (NO. 2018YFA0703900). Jin Ma is supported by US NSF Grants DMS-1908665.


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Rainer Buckdahn. Juan Li. Jin Ma. "A general conditional McKean–Vlasov stochastic differential equation." Ann. Appl. Probab. 33 (3) 2004 - 2023, June 2023. https://doi.org/10.1214/22-AAP1858


Received: 1 September 2021; Revised: 1 June 2022; Published: June 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583663
zbMATH: 07692310
Digital Object Identifier: 10.1214/22-AAP1858

Primary: 34F05 , 35R60 , 60H07 , 60H15 , 60H30

Keywords: Conditional McKean–Vlasov SDEs , Kantorovich–Rubinstein’s duality , Weak solution

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 3 • June 2023
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