The Conditional Poincar\'e Inequality for Filter Stability

TL;DR

This paper introduces the conditional Poincaré inequality as a new tool to establish nonlinear filter stability for ergodic Markov processes, linking stochastic control and stability analysis.

Contribution

It presents the conditional Poincaré inequality and demonstrates its effectiveness in proving filter stability through a novel duality approach.

Findings

Conditional PI yields filter stability

Duality transforms filtering into stochastic control

Comparison between stochastic process stability and filter stability

Abstract

This paper is concerned with the problem of nonlinear filter stability of ergodic Markov processes. The main contribution is the conditional Poincar\'e inequality (PI), which is shown to yield filter stability. The proof is based upon a recently discovered duality which is used to transform the nonlinear filtering problem into a stochastic optimal control problem for a backward stochastic differential equation (BSDE). Based on these dual formalisms, a comparison is drawn between the stochastic stability of a Markov process and the filter stability. The latter relies on the conditional PI described in this paper, whereas the former relies on the standard form of PI.