Generic well-posedness for a family of quadratic BSDE systems
Abstract
We study a two-parameter family of quadratic BSDE systems of the form given by Jackson (2023) in his open questions on non-Markovian solvability. Fan, Hu, and Tang (2025) established well-posedness for every bounded terminal datum when 1/α+1/β=1. For every α,β>0 and M>0, we establish existence and uniqueness in 𝒮∞×BMO for a dense Gδ subset of the space of terminal data satisfying |ξi|≤ M a.s., i=1,2, equipped with the topology of convergence in probability. We also obtain well-posedness for every terminal datum taking at most two values. Furthermore, we show that every bounded terminal problem can be reduced to one with a three-valued terminal datum while preserving its solution set. However, for every positive parameter pair with 1/α+1/β≠1, we construct a three-valued terminal datum admitting at least two distinct solutions in 𝒮∞×BMO. In particular, uniqueness can fail in Jackson's stochastic-game example α=β=1.
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BibTeX
@misc{wang2026generic,
title = {{Generic well-posedness for a family of quadratic BSDE systems}},
author = {Siyi Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.29762},
eprint = {2609.29762},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29762},
}
RIS
TY - UNPB
TI - Generic well-posedness for a family of quadratic BSDE systems
AU - Siyi Wang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.29762
UR - https://arxiv.org/abs/2609.29762
AB - We study a two-parameter family of quadratic BSDE systems of the form given by Jackson (2023) in his open questions on non-Markovian solvability. Fan, Hu, and Tang (2025) established well-posedness for every bounded terminal datum when 1/α+1/β=1. For every α,β>0 and M>0, we establish existence and uniqueness in 𝒮^∞×BMO for a dense G_δ subset of the space of terminal data satisfying |ξ_i|≤ M a.s., i=1,2, equipped with the topology of convergence in probability. We also obtain well-posedness for every terminal datum taking at most two values. Furthermore, we show that every bounded terminal problem can be reduced to one with a three-valued terminal datum while preserving its solution set. However, for every positive parameter pair with 1/α+1/β≠1, we construct a three-valued terminal datum admitting at least two distinct solutions in 𝒮^∞×BMO. In particular, uniqueness can fail in Jackson's stochastic-game example α=β=1.
ER -