Generic well-posedness for a family of quadratic BSDE systems

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 arbitrary bounded terminal data when $1/α+1/β=1$. For every $α,β>0$ and $M>0$, we prove that the space of terminal data with components bounded by $M$, equipped with convergence in probability, contains a dense $G_δ$ set on which the system has a unique solution in $\mathcal S^\infty\times\mathrm{BMO}$. This gives generic well-posedness in the sense of Baire category for all positive parameters, including Jackson's stochastic-game example $α=β=1$. The solution map is continuous on this set in $\mathcal S^p\times\mathcal H^p$ for every $1\le p<\infty$. The proof combines uniform BMO estimates, stability on a dense class of solvable terminal data, and Baire's theorem. We also establish well-posedness for arbitrary two-valued terminal data and show that solvability for all bounded terminal data is equivalent to solvability for all three-valued terminal data.