On some filtration procedure for jump Markov process observed in white Gaussian noise
The following filtering problem is discussed: \[ Y(t)=\int^ t_ 0 X(s)ds+\sigma W(t), \] is the observed process, \(X(t)\) is a Markov process with the states 0 and 1, and transition probability \(\left({1- \lambda\atop\mu}{\lambda\atop 1-\mu}\right)\), \(W(t)\) is a standard Wiener process, and \(\sigma\) is a constant. Let \({\mathcal F}^ t_ 0=\sigma\{Y(s),\;0\leq s\leq t\}\), \(\pi(t)=P[X(t)=1\mid{\mathcal F}^ t_ 0]\). The optimal estimator is expressed by \[ \widehat X(t)=\begin{cases} 1 & \text{if } Z(t)\geq 0,\\ 0 & \text{if } Z(t)<0,\end{cases} \] where \(Z(t)=\ln[\pi(t)/(1-\pi(t))]\) satisfies \[ dZ(t)=\sigma^{-2} (dY(t)- 2^{-1} dt)+\bigl(\lambda-\mu+\lambda e^{-Z(t)}- \mu e^{Z(t)}\bigr) dt. \] The authors suggest a new estimator: \[ \overline X(t)=\begin{cases} 1 & \text{if } Z_ 0(t)\geq 0,\\0 & \text{if } Z_ 0(t)<0,\end{cases} \] where \(Z_ 0(t)\) is the solution of \[ dZ_ 0(t)= \sigma^{-2} \bigl(dY(t)- 2^{-1} dt\bigr), \] and show its asymptotic efficiency as \(\sigma\to 0\). It is worth noting that \(\overline X(t)\) does not depend on \(\lambda\) and \(\mu\).
scientific article; zbMATH DE number 166859