标题： 动量空间方法在随机滤波渐近展开中的应用 显示英文标题

标题： Momentum-Space Approach to Asymptotic Expansion for Stochastic Filtering

摘要： 本文发展了一种在动量空间中的渐近展开技术用于随机滤波。 结果表明，傅里叶变换结合非线性项的多项式函数逼近可给出相关条件分布的常微分方程（ODE）系统的封闭递归系统。 由于ODE系统的简单性，高阶计算可以轻松完成。 此外，通过使用更新初值条件的小子周期顺序求解ODE，使得以数值高效的方式实现渐近展开的子步方法成为可能。 发现这种方法在其他情况下近似效果极差的问题上显著提高了性能。 该方法有望为更现实的含未观测参数的金融建模以及涉及非线性测度值过程的问题提供有用的工具。 显示英文摘要

摘要： This paper develops an asymptotic expansion technique in momentum space for stochastic filtering. It is shown that Fourier transformation combined with a polynomial-function approximation of the nonlinear terms gives a closed recursive system of ordinary differential equations (ODEs) for the relevant conditional distribution. Thanks to the simplicity of the ODE system, higher order calculation can be performed easily. Furthermore, solving ODEs sequentially with small sub-periods with updated initial conditions makes it possible to implement a substepping method for asymptotic expansion in a numerically efficient way. This is found to improve the performance significantly where otherwise the approximation fails badly. The method is expected to provide a useful tool for more realistic financial modeling with unobserved parameters, and also for problems involving nonlinear measure-valued processes.