An elementary approach to filtering in systems with fractional Brownian observation noise (Q2769687)

The authors consider the optimal filtering problem: Find the best mean-square estimator for the random signal \(X= (X_t, t\in [0,T])\) which is observed through a possibly nonlinear channel \(Y= (Y_t, t\in [0,T])\), which is a stochastic process defined as follows NEWLINE\[NEWLINEY_t= \xi+\int^t_0A(s,X_s)ds+ \int^t_0 B(s) dW_s,\qquad t\in [0,T].NEWLINE\]NEWLINE Here \(W= (W_t, t\in [0,T])\) is a fractional Brownion motion (fBm) with Hurst parameter \(H\in ({1\over 2},1)\), \(A\) and \(B\) are given functions satisfying some regularity conditions and the initial value \(\xi\) is a random variable independent of \(W\). Hence the model studied in this paper can be regarded as an extension of the classical Kalman-Bucy scheme (this is the case \(H={1\over 2}\)). The classical tools cannot be used for models involving fBm. New ideas and techniques are necessary.NEWLINENEWLINENEWLINEFirst the authors establish an auxiliary result showing how to transform theian martingale. (Recall the well-known fact that the fBm is not a semimartingaleot a semimartingale.) Then they derive Girsanov type formula and describe the innovation process for the model \((X,Y)\). Based on these results they find the optimal filter and the conditional variance of the filtering error. Special attention is paid to the case when the dynamics of the process \((X,Y)\) is linear. In this case the optimal filter and its variance are described explicitly and elegantly.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].