W-BSDEs with Common Noise: Well-posedness, Characterization, and Consequences
Abstract
We study a notion of backward stochastic differential equation on the Wasserstein space, called a W-BSDE. This work extends to the common noise setting recent results of the first author. The equation is formulated in Markovian form along admissible flows of conditional laws generated by McKean-Vlasov dynamics with common noise. A solution is a pair of fields (Y,Z), where Y is a functional on the space of probability measures and Z is the corresponding intrinsic Wasserstein gradient. Our main result shows that the well-posedness of the W-BSDE can be reduced to a characteristic equation for a field p. Whenever this equation admits a suitable solution, we construct a W-BSDE solution and identify Z with both the characteristic field p and the intrinsic gradient of Y. This gives an existence result for the W-BSDE and, at the same time, a probabilistic construction of a functional on the Wasserstein space together with its intrinsic gradient. We then develop several consequences of the theory. We obtain verification principles for McKean-Vlasov control problems and for mean field games with common noise, without relying on convexity or separability assumptions. We also show that the characteristic field admits a dynamic representation through a McKean-Vlasov forward-backward SDE. The first component Y solves the associated semilinear PDE on the space of probability measures in the viscosity sense, including the second-order term induced by common noise. Under additional smoothness, Y is a classical solution. Finally, we prove that W-BSDEs arise as intrinsic mean-field limits of classical finite-dimensional BSDEs written on empirical measures, with convergence of both the value component and the rescaled martingale integrands.
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BibTeX
@misc{djete2026wbsdes,
title = {{W-BSDEs with Common Noise: Well-posedness, Characterization, and Consequences}},
author = {Mao Fabrice Djete and Mattia Martini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06593},
eprint = {2610.06593},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06593},
}
RIS
TY - UNPB
TI - W-BSDEs with Common Noise: Well-posedness, Characterization, and Consequences
AU - Mao Fabrice Djete
AU - Mattia Martini
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06593
UR - https://arxiv.org/abs/2610.06593
AB - We study a notion of backward stochastic differential equation on the Wasserstein space, called a W-BSDE. This work extends to the common noise setting recent results of the first author. The equation is formulated in Markovian form along admissible flows of conditional laws generated by McKean-Vlasov dynamics with common noise. A solution is a pair of fields (Y,Z), where Y is a functional on the space of probability measures and Z is the corresponding intrinsic Wasserstein gradient. Our main result shows that the well-posedness of the W-BSDE can be reduced to a characteristic equation for a field p. Whenever this equation admits a suitable solution, we construct a W-BSDE solution and identify Z with both the characteristic field p and the intrinsic gradient of Y. This gives an existence result for the W-BSDE and, at the same time, a probabilistic construction of a functional on the Wasserstein space together with its intrinsic gradient. We then develop several consequences of the theory. We obtain verification principles for McKean-Vlasov control problems and for mean field games with common noise, without relying on convexity or separability assumptions. We also show that the characteristic field admits a dynamic representation through a McKean-Vlasov forward-backward SDE. The first component Y solves the associated semilinear PDE on the space of probability measures in the viscosity sense, including the second-order term induced by common noise. Under additional smoothness, Y is a classical solution. Finally, we prove that W-BSDEs arise as intrinsic mean-field limits of classical finite-dimensional BSDEs written on empirical measures, with convergence of both the value component and the rescaled martingale integrands.
ER -