Sharp propagation of chaos for mean-field backward stochastic differential equations
Abstract
We study propagation of chaos for decoupled mean-field forward-backward stochastic differential equations whose generators depend on the empirical laws of the forward states, backward values and diagonal martingale integrands. Under monotonicity and Lipschitz assumptions, synchronous coupling gives quantitative estimates, including an m-particle squared Wasserstein bound of order m/n for a system of n particles interacting through finitely many statistics. In the Markovian setting, assuming a sufficiently regular classical decoupling field, we obtain two sharp refinements. For constant invertible diffusion and first-order cancellation of the field's measure dependence along the limiting law flow, the squared Wasserstein error is of order m2/n2, on continuous-path space for the values and on L2 for the diagonal integrands. Without imposing this cancellation, smooth weak errors have order n−1 for every fixed marginal, allowing variable and possibly degenerate diffusion. The weak estimate is uniform on a fixed time interval for the values and integrated in time for the integrands. The argument compares the interacting BSDE with an empirical evaluation of the decoupling field, retaining the full martingale representation and controlling feedback through both backward laws. It yields a joint-path Wasserstein transfer bound with intrinsic squared error m/n2, off-diagonal integrand estimates, and a weak-error transfer principle with additive error n−1. Explicit models with feedback through both backward laws verify the cancellation assumptions. Examples distinguish the intrinsic backward error from the forward law error and establish matching lower bounds for each backward component.
Related
- Backward and reflected backward stochastic differential equations under initial enlargement of Brownian filtrations
- Backward doubly stochastic differential equations with or without reflection under weak conditions
- Backward stochastic partial differential equations with unbounded random coefficients
- Conditional Reflected Backward Stochastic Differential Equations
- Multidimensional Infinite-Horizon Markovian Backward Stochastic Differential Equations with Continuous Coefficients and Nonzero-Sum Differential Games
BibTeX
@misc{gao2026sharp,
title = {{Sharp propagation of chaos for mean-field backward stochastic differential equations}},
author = {Shuxian Gao and Ying Hu and Jiaqiang Wen},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15798},
eprint = {2609.15798},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05751209v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Sharp propagation of chaos for mean-field backward stochastic differential equations
AU - Shuxian Gao
AU - Ying Hu
AU - Jiaqiang Wen
PY - 2026
DA - 2026/09/14
PB - HAL
DO - 10.48550/arxiv.2609.15798
UR - https://hal.science/hal-05751209v1
AB - We study propagation of chaos for decoupled mean-field forward-backward stochastic differential equations whose generators depend on the empirical laws of the forward states, backward values and diagonal martingale integrands. Under monotonicity and Lipschitz assumptions, synchronous coupling gives quantitative estimates, including an m-particle squared Wasserstein bound of order m/n for a system of n particles interacting through finitely many statistics. In the Markovian setting, assuming a sufficiently regular classical decoupling field, we obtain two sharp refinements. For constant invertible diffusion and first-order cancellation of the field's measure dependence along the limiting law flow, the squared Wasserstein error is of order m^2/n^2, on continuous-path space for the values and on L^2 for the diagonal integrands. Without imposing this cancellation, smooth weak errors have order n^−1 for every fixed marginal, allowing variable and possibly degenerate diffusion. The weak estimate is uniform on a fixed time interval for the values and integrated in time for the integrands. The argument compares the interacting BSDE with an empirical evaluation of the decoupling field, retaining the full martingale representation and controlling feedback through both backward laws. It yields a joint-path Wasserstein transfer bound with intrinsic squared error m/n^2, off-diagonal integrand estimates, and a weak-error transfer principle with additive error n^−1. Explicit models with feedback through both backward laws verify the cancellation assumptions. Examples distinguish the intrinsic backward error from the forward law error and establish matching lower bounds for each backward component.
ER -