发表机构
Southern University of Science and Technology; Peking University（南方科技大学; 北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对具有无界随机系数的多维非线性滤波模型，通过局部化熵论证和停时Zakai方程推导出全局滤波方程，并利用对偶倒向随机偏微分方程证明解的唯一性，最后在W_1距离下建立稳定性结果。
AI 中文摘要
我们研究了一个多维非线性滤波模型，其系数依赖于给定的观测自适应可料过程，且观测漂移可能关于状态和随机输入均线性增长。由于观测漂移的无界性，全局参考测度不可用。为克服这一障碍，我们采用局部化熵论证，证明在每个控制能量停时水平上，停时似然是一个一致可积鞅。由此推导出停时Zakai方程，进而得到停时滤波方程。通过去局部化建立全局滤波方程。停时Zakai方程解的唯一性通过一个对偶倒向随机偏微分方程获得。该唯一性进而通过停时方程传播到全局滤波方程的唯一性。最后，在$W_1$距离下建立了稳定性结果。
英文摘要
We study a multidimensional nonlinear filtering model whose coefficients depend on a given observation-adapted predictable process and whose observation drift may grow linearly in both the state and the random input. Due to the unboundedness of the observation drift, a global reference measure is not available. To overcome this hurdle, a localized entropy argument is adapted to prove the stopped likelihood to be a uniformly integrable martingale at each control-energy stopping level. The stopped Zakai equation, and hence, the stopped filtering equation is derived. The global filtering equation is then established by de-localization. The uniqueness of the solution to the stopped Zakai equation is obtained by a duality backward stochastic partial differential equation. This uniqueness then propagates to that of the global filtering equation through the stopped ones. Finally, a stability result is established in $W_1$-distance of measures.
