一类二次BSDE系统的泛适定性
Generic well-posedness for a family of quadratic BSDE systems
浏览论文内容
中文总结 AI 辅助
本文证明了一类双参数二次BSDE系统在所有正参数下具有Baire范畴意义上的泛适定性，并建立了解映射的连续性与多值终端数据的适定性等价关系。
中文摘要 AI 辅助
我们研究由Jackson（2023）在其关于非马尔可夫可解性的开放问题中给出形式的一族双参数二次BSDE系统。Fan、Hu和Tang（2025）在$1/\alpha+1/\beta=1$时对任意有界终端数据建立了适定性。对于每个$\alpha,\beta>0$和$M>0$，我们证明：由分量以$M$为界的终端数据构成的空间（配备依概率收敛）包含一个稠密$G_\delta$集合，在该集合上系统在$\mathcal S^\infty\times\mathrm{BMO}$中存在唯一解。这给出了所有正参数下Baire范畴意义上的泛适定性，包括Jackson的随机博弈例子$\alpha=\beta=1$。解映射在该集合上对每个$1\le p<\infty$在$\mathcal S^p\times\mathcal H^p$中连续。证明结合了均匀BMO估计、可解终端数据稠密类上的稳定性以及Baire定理。我们还对任意二值终端数据建立了适定性，并证明所有有界终端数据的可解性等价于所有三值终端数据的可解性。
英文摘要
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 $\mathcal{S}^{\infty}\times\mathrm{BMO}$ for a dense $G_δ$ subset of the space of terminal data satisfying $|ξ_i|\le M$ 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 $1/α+1/β\ne1$, we construct a three-valued terminal datum admitting at least two distinct solutions in $\mathcal{S}^{\infty}\times\mathrm{BMO}$. In particular, uniqueness can fail in Jackson's stochastic-game example $α=β=1$.
