Conditional McKean-Vlasov control
Abstract
Conditional McKean-Vlasov control problems involve controlling McKean-Vlasov diffusions where the interaction occurs through the law of the state process conditionally on it staying in a domain. Introduced by Lions in his 2016 lectures at the Coll\`ege de France, these problems have notable applications, particularly in systemic risk. We establish well-posedness and provide a general characterization of optimal controls using a new Pontryagin maximum principle in the probabilistic weak formulation. Unlike the classical approach based on forward-backward systems, our results connect the control problem to a generalized McKean-Vlasov backward stochastic differential equation (BSDE). We illustrate our framework with two applications: a version of the Schr\"odinger problem with killing, and a construction of equilibria in potential mean field games via McKean-Vlasov control.
Cite
@article{arxiv.2510.06543,
title = {Conditional McKean-Vlasov control},
author = {René Carmona and Ludovic Tangpi and Kaiwen Zhang},
journal= {arXiv preprint arXiv:2510.06543},
year = {2025}
}