TY - JOUR
T1 - LINEAR FILTERING WITH FRACTIONAL NOISES
T2 - LARGE TIME AND SMALL NOISE ASYMPTOTICS
AU - Afterman, Danielle
AU - Chigansky, Pavel
AU - Kleptsyna, Marina
AU - Marushkevych, Dmytro
N1 - Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics.
PY - 2022
Y1 - 2022
N2 - 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.
AB - 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.
KW - asymptotic analysis
KW - fractional Brownian motion
KW - stochastic filtering
UR - http://www.scopus.com/inward/record.url?scp=85132364467&partnerID=8YFLogxK
U2 - 10.1137/20M1360359
DO - 10.1137/20M1360359
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AN - SCOPUS:85132364467
SN - 0363-0129
VL - 60
SP - 1463
EP - 1487
JO - SIAM Journal on Control and Optimization
JF - SIAM Journal on Control and Optimization
IS - 3
ER -