On filtering over Îto-Volterra observations (Q5932230)

Let \((x(t),y(t))\), \(t\geq 0\), be a partially observed random process defined by a system of Itô-Volterra equations of the form NEWLINE\[NEWLINE\begin{aligned} x (t) &=\int_0^t\big( a_0(t,s)+a(t,s)x(s)\big) ds +\int_0^tb(t,s) dW^2(s),\\ y(t) &=\int_0^t\big(A_0(t,s)+A(t,s)x(s)\big) ds +\int_0^tB(t,s) dW^2(s),\end{aligned}NEWLINE\]NEWLINE driven by independent (multidimensional) Wiener processes \(W^1\) and \(W^2\). The author constructs the Kalman-Bucy filter for the nonobserved component \(x\) over the observed process \(y\), i.e., he derives an equation system satisfied by the expectation \(m(t)=E(x(t)|F^Y_t)\), the correlation function \(P(t)=E\{(x(t)-m(t))(x(t)-m(t))^T|F^Y_t\}\), and a certain additional function \(f(t,s)\) characterizing the deviation of \(m(t)\) from the true state \(x(t)\). Here \(F^Y_t=\sigma\{y_s, s\leq t\}\). The author also constructs the Kalman-Bucy filter for an Itô-Volterra process over discontinuous Itô-Volterra observations and, as a consequence, over discrete observations with delays.