Kalman-Bucy filter and SPDEs with growing lower-order coefficients in $W^{1}_{p}$ spaces without weights

TL;DR

This paper studies divergence form parabolic SPDEs with certain coefficient conditions, providing solutions in Sobolev spaces and extending filtering theory to include Zakai's equation for the Kalman-Bucy filter.

Contribution

It introduces methods to handle SPDEs with growing lower-order coefficients in Sobolev spaces without weights, integrating Zakai's equation into the filtering framework.

Findings

Solutions exist in $W^{1}_{p}$ spaces for the considered SPDEs.

The approach includes Zakai's equation for the Kalman-Bucy filter.

Extended filtering theory to broader coefficient classes.

Abstract

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, $p\geq2$, 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.