A time-fractional Kalman filter

Abstract

We study a linear filtering problem in which the signal process is described by a time-fractional linear stochastic differential equation driven by Brownian motion. We derive a stochastic integral equation for the conditional mean alongside a Riccati--Volterra type integral equation for the mean-square error function. As a core application, we introduce a time-fractional state-estimation framework for modelling learning trajectories in children with developmental dyscalculia.