Momentum-Space Approach to Asymptotic Expansion for Stochastic Filtering

TL;DR

This paper introduces a momentum-space asymptotic expansion method for stochastic filtering that simplifies calculations and enhances numerical efficiency, especially for complex nonlinear problems in finance and measure-valued processes.

Contribution

It presents a novel Fourier-based asymptotic expansion technique that yields a recursive ODE system, enabling efficient higher-order approximations and a substepping method for improved performance.

Findings

The method produces a closed recursive ODE system for filtering.

Substepping with updated initial conditions improves approximation accuracy.

The approach is promising for realistic financial models with unobserved parameters.

Abstract

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.