Kalman-Bucy filter and SPDEs with growing lower-order coefficients in $W_p^1$ spaces without weights

We consider divergence form uniformly parabolic SPDEs with VMO bounded leading coefficients, bounded coefficients in the stochastic part, and possibly growing lower-order coefficients in the deterministic part. We look for solutions which are summable to the $p$th power, $p\ge 2$, with respect to the usual Lebesgue measure along with their first-order derivatives with respect to the spatial variable. Our methods allow us to include Zakai’s equation for the Kalman-Bucy filter into the general filtering theory.