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Linear filtering with fractional Brownian motion. (English) Zbl 0916.93076

The following linear filtering model is studied: \[ d\theta_s = a(s)\theta_s ds + dB_s^h, \qquad d\zeta_s = A(s)\theta_s ds + dW_s , \] with signal \(\theta_s\), observation \(\zeta_s\) and observation noise \(W_s \) (a Brownian motion). The noise generating the signal is a fractional Brownian motion, \(B_t^h\), with Hurst index \(h\).
The authors obtain a Kalman-type system of integral equations for the conditional expectation of the signal \(\theta_t\). They give also an explicit expression for the error in a particular case.

MSC:

93E11 Filtering in stochastic control theory
60G20 Generalized stochastic processes
60G35 Signal detection and filtering (aspects of stochastic processes)
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References:

[1] Lin S.J., Stochastics and Stochastic Reports 55 pp 121– (1995)
[2] Liptser R.S., Statistics of Random Processes (1977) · Zbl 0364.60004
[3] Liptser R.S., Theory of Martingales (1984)
[4] DOI: 10.1137/1010093 · Zbl 0179.47801 · doi:10.1137/1010093
[5] Rozovskii B.L., Evolutionary Stochastic Systems, Linear Theory with Applications to the Statistics of Random Processes (1983)
[6] Samorodnitsky G., Stable Non-Gaussian Random Processes (1994) · Zbl 0925.60027
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