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 It\^{o} representations of their limiting forms as the approximation parameter $\delta \rightarrow 0$. 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.

翻译：在过去几十年里,提出了各种粒子过滤器,其共同特征是更新步骤受某种类型的控制法管辖。这一特征使其具有吸引力,取代传统的连续的Monte Carlo,后者因重量下降而与状态层面相比不相称。本条款提议了一个统一框架,允许系统地从离散时间和连续时间观察案例中提取这些过滤器的麦肯-弗拉索夫表示法,从Crisan & Xiong(2010年)和Clark & Crisan(2005年)所考虑的数据的平稳近似中得到启发,我们考虑了文献中提议的三个过滤器,并利用这个框架得出其限制形式的表示法作为近似参数 $\delta\rightrowr 0美元。所有过滤器都需要用$\mathb{R ⁇ d}美元定义的Poisson方程式的解决方案,其存在和独特性可能是一个非技术性问题。我们还在信号-观察系统中建立了各种条件,以确保一个过滤器产生的加权Poisson方程式的正确性。