The Conditional Poincaré Inequality for Filter Stability

Jin Won Kim, Prashant G. Mehta, Sean Meyn

Published 2021-03-26Version 1

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 result whereby the nonlinear filtering problem is cast as a stochastic optimal control problem for a backward stochastic differential equation (BSDE). A comparison is made between the stochastic stability of a Markov process and the filter stability. The former is based upon assuming the standard form of PI whereas the latter relies on the conditional PI introduced in this paper.