Backward and forward filtering under the weak H\"ormander condition

TL;DR

This paper develops forward and backward filtering equations for degenerate diffusions under the weak H"ormander condition, using H"older theory for SPDEs, and provides results on the existence, regularity, and estimates of the filtering density.

Contribution

It introduces a novel approach based on H"older theory to derive filtering equations for degenerate diffusions without relying on general filtering theory results.

Findings

Derived filtering equations for degenerate diffusions.

Established existence and regularity of the filtering density.

Provided estimates for the filtering density.

Abstract

We derive the forward and backward filtering equations for a class of degenerate partially observable diffusions, satisfying the weak H\"ormander condition. Our approach is based on the H\"older theory for degenerate SPDEs that allows to pursue the direct approaches proposed by N. V. Krylov and A. Zatezalo, and A. Yu. Veretennikov, avoiding the use of general results from filtering theory. As a by-product we also provide existence, regularity and estimates for the filtering density.