Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear State Model with State-Independent Diffusion Matrix

TL;DR

This paper presents a universal filtering approach for affine, linear state models with state-independent diffusion, enabling solution via linear algebra without differential equations, suitable for real-time, high-dimensional applications.

Contribution

It introduces a method to solve the filtering problem using linear algebra techniques for a specific class of models, avoiding differential equations and allowing arbitrary initial distributions.

Findings

Solution obtained using linear algebra techniques

No differential equations needed for the specified model

Suitable for real-time implementation in high-dimensional problems

Abstract

The continuous-discrete filtering problem requires the solution of a partial differential equation known as the Fokker-Planck-Kolmogorov forward equation (FPKfe). In this paper, it is pointed out that for a state model with an affine, linear drift and state-independent diffusion matrix the fundamental solution can be obtained using only linear algebra techniques. In particular, no differential equations need to be solved. Furthermore, there are no restrictions on the size of the time step size, or on the measurement model. Also discussed are important computational aspects that are crucial for potential real-time implementation for higher-dimensional problems. The solution is universal in the sense that the initial distribution may be arbitrary.