Stability of the optimal filter in a hidden Markov model with multiplicative noise

TL;DR

This paper demonstrates the stability of the optimal filter in a hidden Markov model with multiplicative noise, providing explicit rates and showing stability independence from the signal's ergodic properties.

Contribution

It establishes the stability of the optimal filter under general initial conditions in models with multiplicative noise, with explicit convergence rates.

Findings

Stability of the optimal filter is proven in total variation and L^p norms.

Stability results are independent of the ergodic behavior of the signal.

Explicit rates of convergence are derived.

Abstract

We consider a hidden Markov model with multiplicative noise emerging from studies of software reliability. We show the stability of the optimal filter with respect to general initial conditions in the total variation- and $L^p$-norm and deduce explicit rates. Remarkably, stability turns out to be independent of the ergodic behavior of the signal.