Stability of Optimal Filter Higher-Order Derivatives

TL;DR

This paper investigates the existence, stability, and ergodic properties of higher-order derivatives of the optimal filter in state-space models, crucial for online parameter estimation methods.

Contribution

It establishes the existence, exponential forgetting, and geometric ergodicity of higher-order derivatives of the optimal filter under mild conditions.

Findings

Higher-order derivatives exist and are stable.

Derivatives forget initial conditions exponentially.

Derivatives are geometrically ergodic.

Abstract

In many scenarios, a state-space model depends on a parameter which needs to be inferred from data. Using stochastic gradient search and the optimal filter (first-order) derivative, the parameter can be estimated online. To analyze the asymptotic behavior of online methods for parameter estimation in non-linear state-space models, it is necessary to establish results on the existence and stability of the optimal filter higher-order derivatives. The existence and stability properties of these derivatives are studied here. We show that the optimal filter higher-order derivatives exist and forget initial conditions exponentially fast. We also show that the optimal filter higher-order derivatives are geometrically ergodic. The obtained results hold under (relatively) mild conditions and apply to state-space models met in practice.