Series Expansion Approximations of Brownian Motion for Non-Linear Kalman Filtering of Diffusion Processes

TL;DR

This paper introduces a novel sigma-point method that uses a Fourier series expansion of Brownian motion to improve continuous-discrete filtering of diffusion processes, especially in highly nonlinear scenarios.

Contribution

The paper proposes a new approach that decomposes Brownian motion into Fourier series, enabling more efficient and accurate filtering with fewer computations compared to existing methods.

Findings

The method effectively approximates Brownian motion with fewer Fourier coefficients.

It outperforms traditional discretization-based filters in highly nonlinear filtering problems.

The approach reduces information loss during iterative sigma-point transformations.

Abstract

In this paper, we describe a novel application of sigma-point methods to continuous-discrete filtering. In principle, the nonlinear continuous- discrete filtering problem can be solved exactly. In practice, the solution contains terms that are computationally intractible. Assumed density filtering methods attempt to match statistics of the filtering distribution to some set of more tractible probability distributions. We describe a novel method that decomposes the Brownian motion driving the signal in a generalised Fourier series, which is truncated after a number of terms. This approximation to Brownian can be described using a relatively small number of Fourier coefficients, and allows us to compute statistics of the filtering distribution with a single application of a sigma-point method. Assumed density filters that exist in the literature usually rely on discretisation of the signal dynamics followed by iterated application of a sigma point transform (or a limiting case thereof). Iterating the transform in this manner can lead to loss of information about the filtering distri- bution in highly nonlinear settings. We demonstrate that our method is better equipped to cope with such problems.