LINEAR FILTERING WITH FRACTIONAL NOISES: LARGE TIME AND SMALL NOISE ASYMPTOTICS

Danielle Afterman, Pavel Chigansky, Marina Kleptsyna, Dmytro Marushkevych

Research output: Contribution to journalArticlepeer-review

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 signalto-noise regime. Closed form expressions are derived in a number of important cases.

Original languageAmerican English
Pages (from-to)1463-1487
Number of pages25
JournalSIAM Journal on Control and Optimization
Volume60
Issue number3
DOIs
StatePublished - 2022

Bibliographical note

Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics.

Keywords

  • asymptotic analysis
  • fractional Brownian motion
  • stochastic filtering

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