Non-Linear Continuous-Discrete Smoothing by Basis Function Expansions of Brownian Motion

Abstract

This paper is concerned with inferring the state of a Itô stochastic differential equation (SDE) from noisy discrete-time measurements. The problem is approached by considering basis function expansions of Brownian motion, that as a consequence give approximations to the underlying stochastic differential equation in terms of an ordinary differential equation with random coefficients. This allows for representing the latent process at the measurement points as a discrete time system with a non-linear transformation of the previous state and a noise term. The smoothing problem can then be solved by sigma-point or Taylor series approximations of this non-linear function, implementations of which are detailed. Furthermore, a method for interpolating the smoothing solution between measurement instances is developed. The developed methods are compared to the Type III smoother in simulation examples involving (i) hyperbolic tangent drift and (ii) the Lorenz 63 system where the present method is found to be better at reconstructing the smoothing solution at the measurement points, while the interpolation scheme between measurement instances appear to suffer from edge effects, serving as an invitation to future research.