Linear filtering with fractional Brownian motion in the signal and observation processes (Q1305824)

The authors consider the linear problem with the signal \(\theta _{t}\) and the observation \(\xi _{t}\) defined by the linear equations \[ \theta _{t}=\int_{0}^{t}a(s)\theta _{s}ds+B_{t}^{h},\quad \xi _{t}=\int_{0}^{t}A(s)\theta _{s}ds+W_{t}+B_{t}^{h}, \] where \(B_{t}^{h}\) is a fractional Brownian motion with Hurst index \(h\in (3/4,1)\) and \(W_{t}\) is a standard Wiener process. For \(\hat{\theta}_{t}=E(\theta _{t}|\xi _{s},0\leq s\leq t)\) they obtain the expression \(\hat{\theta}_{t}=\int_{0}^{t}\Phi (t,s)d\xi _{s}\) where \(\Phi \) can be found from an integral equation derived in the paper.