论文标题
非线性过滤中的McKean-Vlasov SDE
McKean-Vlasov SDEs in nonlinear filtering
论文作者
论文摘要
在过去的几十年中，已经提出了各种粒子过滤器，其更新步骤由一种控制定律控制的共同特征。此功能使它们成为传统顺序蒙特卡洛的有吸引力替代品，由于重量退化而导致状态尺寸缩小范围很差。本文提出了一个统一的框架，该框架可以在离散的时间和连续的时间观察案例中系统地得出这些过滤器的McKean-Vlasov表示，从而从Crisan＆Xiong（2010）和Clark＆Crisan（2005）中考虑的数据的平稳近似中获得了灵感。我们认为文献中提出的三个过滤器，并使用此框架将其限制形式的ITô表示为近似参数$δ\ rightarrow 0 $。所有过滤器都需要在$ \ mathbb {r}^{d} $上定义的泊松方程的解决方案，而解决方案的存在和独特性可能是一个非平凡的问题。我们还在信号观察系统上建立条件，以确保其中一个过滤器中产生的加权泊松方程的拟合度。
Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in Crisan & Xiong (2010) and Clark & Crisan (2005). We consider three filters that have been proposed in the literature and use this framework to derive Itô representations of their limiting forms as the approximation parameter $δ\rightarrow 0$. All filters require the solution of a Poisson equation defined on $\mathbb{R}^{d}$, for which existence and uniqueness of solutions can be a non-trivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.
