Linear filtering with fractional noises: large time and small noise asymptotics

TL;DR

This paper extends the classical Kalman-Bucy filtering framework to systems driven by fractional Brownian motions, developing asymptotic analysis methods to understand steady-state errors and their scaling in high noise regimes.

Contribution

It introduces a new asymptotic analysis method for fractional noise-driven filtering equations and derives explicit steady-state error expressions.

Findings

Existence of steady-state error limit established

Asymptotic scaling of error in high signal-to-noise regime derived

Closed-form expressions obtained for key cases

Abstract

The classical state-space approach to optimal estimation of stochastic processes is efficient when the driving noises are generated by martingales. In particular, the weight function of the optimal linear filter, which solves a complicated operator equation in general, simplifies to the Riccati ordinary differential equation in the martingale case. This reduction lies in the foundations of the Kalman-Bucy approach to linear optimal filtering. In this paper we consider a basic Kalman-Bucy model with noises, generated by independent fractional Brownian motions, and develop a new method of asymptotic analysis of the integro-differential filtering equation arising in this case. We establish existence of the steady-state error limit and find its asymptotic scaling in the high signal-to-noise regime. Closed form expressions are derived in a number of important cases.