Hypoellipticity for filtering problems of partially observable diffusion processes

TL;DR

This paper proves that under certain conditions, the filtering density for partially observable diffusion processes is infinitely differentiable and can be represented via a smooth kernel, even with minimal regularity assumptions on coefficients.

Contribution

It establishes the existence and smoothness of the filtering kernel under H"ormander's conditions, extending results to coefficients that are only measurable in time.

Findings

Filtering density is infinitely differentiable under H"ormander's conditions.

The filtering kernel exists and is smooth.

Results apply even when coefficients are only measurable in time.

Abstract

We prove that under H\"ormander's type conditions on the coefficients of the unobservable component of a partially observable diffusion process the filtering density is infinitely differentiable and can be represented as the integral of an infinitely differentiable kernel against the prior initial distribution. These results are derived from more general results obtained for SPDEs. One the main novelty of the paper is the existence and smoothness of the kernel, another that we allow the coefficients of our partially observable process to be just measurable with respect to the time variable.