发表机构
Southern University of Science and Technology; Peking University; The Hong Kong Polytechnic University（南方科技大学; 北京大学; 香港理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究具有无界随机系数的线性倒向随机偏微分方程，通过条件参数估计等方法，在加权Sobolev类中证明解的存在唯一性。
AI 中文摘要
我们研究一类具有无界观测适应随机系数的线性倒向随机偏微分方程，其动机来自非线性滤波的对偶方法。系数通过一个控制以几乎必然有限的时间积分平方范数，对观测历史进行可预测依赖。漂移、扩散以及乘在鞅被积项上的系数可以在空间上线性增长。扩散矩阵可以在空间上二次增长，并且允许退化。在有界正阶空间导数和有界光滑观测可测终端数据的条件下，我们在一个多项式加权的Sobolev类中，证明了解的存在唯一性，直至控制能量停时。证明结合了条件参数估计、似然的熵界、参考概率下的稳定性以及随机流重构。
英文摘要
We study a linear backward stochastic partial differential equation with unbounded observation-adapted random coefficients, motivated by the duality approach to nonlinear filtering. The coefficients depend predictably on the observation history through a control with almost surely finite time-integrated squared norm. The drift, diffusion, and coefficient multiplying the martingale integrand may grow linearly in space. The diffusion matrix may grow quadratically in space and is allowed to be degenerate. Under bounded positive-order spatial derivatives and bounded smooth observation-measurable terminal data, we prove existence and uniqueness in a polynomially weighted Sobolev class up to control-energy stopping times. The proof combines conditional parameter estimates, entropy bounds for likelihoods, stability under the reference probability, and stochastic-flow reconstruction.
