论文标题
最佳投影过滤器
Optimal Projection Filters
论文作者
论文摘要
我们介绍了在阿姆斯特朗（Armstrong）开发的两个随机微分方程（SDE）投影的新概念，Brigo e Rossi Ferrucci（2019，2018）：ITO-VECTOR和ITO-JET投影。这允许使用微分几何技术系统和最佳地开发高维SDE的低维近似值。我们的新预测是基于最佳参数，并在均方根意义上产生了针对原始SDE的明确定义的``最佳''近似值。我们还表明，较早的Stratonovich投影满足了最佳标准，该标准比新预测所满足的标准更加临时和自然。作为应用程序，我们考虑使用hellinger或$ l^2 $直接指标以及密度空间上的相关信息几何结构近似于给定密度的非线性过滤问题的解决方案。 Stratonovich的投影产生了在Brigo，Hanzon和Le Gland（1998，1999）中研究的投影过滤器，而新的预测导致了最佳投影过滤器。最佳投影过滤器是在Armstrong，Brigo E Rossi Ferrucci（2019）中引入的，其中给出了高斯案例的数值示例，并将其与更传统的非线性过滤器进行了比较。
We present the two new notions of projection of a stochastic differential equation (SDE) onto a submanifold, as developed in Armstrong, Brigo e Rossi Ferrucci (2019, 2018): the Ito-vector and Ito-jet projections. This allows one to systematically and optimally develop low dimensional approximations to high dimensional SDEs using differential geometric techniques. Our new projections are based on optimality arguments and yield a well-defined ``optimal'' approximation to the original SDE in the mean-square sense. We also show that the earlier Stratonovich projection satisfies an optimality criterion that is more ad hoc and less natural than the criteria satisfied by the new projections. As an application, we consider approximating the solution of the non-linear filtering problem within a given manifold of densities, using either the Hellinger or $L^2$ direct metrics and related Information Geometry structures on the space of densities. The Stratonovich projection had yielded the projection filters studied in Brigo, Hanzon and Le Gland (1998, 1999), while the new projections lead to the optimal projection filters. The optimal projection filters have been introduced in Armstrong, Brigo e Rossi Ferrucci (2019), where numerical examples for the Gaussian case are given and where they are compared to more traditional nonlinear filters.
