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

In this paper, we describe a novel application of sigma-point methods to continuous-discrete filtering. The nonlinear continuous-discrete filtering problem is often computationally intractable to solve. Assumed density filtering methods attempt to match statistics of the filtering distribution to some set of more tractable probability distributions. Filters such as these are usually decompose the problem into two sub-problems. The first of these is a prediction step, in which one uses the known dynamics of the signal to predict its state at time $t_{k+1}$ given observations up to time $t_{k}$. In the second step, one updates the prediction upon arrival of the observation at time $t_{k+1}$ . The aim of this paper is to describe a novel method that improves the prediction step. We decompose the Brownian motion driving the signal in a generalised Fourier series, which is truncated after a number of terms. This approximation to Brownian motion 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 distribution in highly non-linear settings. We demonstrate that our method is better equipped to cope with such problems.