1. LINEAR FILTERING WITH FRACTIONAL NOISES: LARGE TIME AND SMALL NOISE ASYMPTOTICS.
- Author
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AFTERMAN, DANIELLE, CHIGANSKY, PAVEL, KLEPTSYNA, MARINA, and MARUSHKEVYCH, DMYTRO
- Subjects
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BROWNIAN motion , *ORDINARY differential equations , *OPERATOR equations , *KALMAN filtering , *STOCHASTIC processes , *INTEGRO-differential equations , *NOISE , *MARTINGALES (Mathematics) - 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. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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