标题： 动量空间方法用于随机滤波及其他问题的渐近展开 显示英文标题

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

摘要： 本文开发了一种动量空间中的渐近展开技术用于随机滤波。 结果显示，傅里叶变换结合非线性项的多项式函数逼近可以为相关的条件分布得到一个闭合的递归常微分方程组（ODE）。 由于ODE系统的简单性，可以轻松进行高阶计算。 此外，通过使用更新的初始条件依次求解小子时间段的ODE，可以以数值高效的方式实现{\it 亚步方法}的渐近展开。 发现在其他情况下近似效果很差的地方，这种方法显著提高了性能。 该方法有望为具有未观测参数的更现实的金融建模以及涉及非线性测度值过程的问题提供一个有用的工具。 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 {\it 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 an useful tool for more realistic financial modeling with unobserved parameters, and also for problems involving nonlinear measure-valued processes.