SPDEs in divergence form with VMO coefficients and filtering theory of partially observable diffusion processes with Lipschitz coefficients

TL;DR

This paper investigates the smoothness of filtering densities for partially observable diffusion processes with Lipschitz coefficients, using divergence form equations to enhance understanding in stochastic filtering theory.

Contribution

It introduces a novel approach by rewriting filtering equations in divergence form, facilitating analysis of smoothness in $L_{p}$ spaces under Lipschitz conditions.

Findings

Established $L_{p}$ smoothness of filtering densities.

Rewrote filtering equations in divergence form for better analytical tractability.

Provided new insights into filtering theory for diffusion processes.

Abstract

We present several results on the smoothness in $L_{p}$ sense of filtering densities under the Lipschitz continuity assumption on the coefficients of a partially observable diffusion processes. We obtain them by rewriting in divergence form filtering equation which are usually considered in terms of formally adjoint to operators in nondivergence form.