Convergence of the extended Kalman filter with small and state-dependent noise

TL;DR

This work addresses nonlinear filtering with state-dependent observation noise and small diffusion, focusing on the extended Kalman filter (EKF) convergence. It generalizes two key results from Picard to a setting where the observation diffusion can depend on the state, proving that when the observation drift is strongly injective and the drifts become nearly linear as $\varepsilon\to 0$, the filtering error scales as $O(\sqrt{\varepsilon})$, and that the initial filtering error is forgotten exponentially fast under a detectability condition. The paper derives the EKF in this general setting, including the gain $G(t)$ and the Riccati-type covariance dynamics $Q(t)$, and provides a rigorous error-analysis framework using notions like almost-linearity, strong injectivity, and $K$-stability. An illustrative motivating example from epidemic modeling (SI$^{\pm}$S) demonstrates the model's applicability and confirms the plausibility of the assumptions in a concrete context. The results offer practically relevant guarantees for EKF performance in small-noise regimes with state-dependent observation noise, with implications for real-time nonlinear state estimation in engineering and epidemiology.

Abstract

Nonlinear filtering problems are encountered in many applications, and one solution approach is the extended Kalman filter, which is not always convergent. Therefore, it is crucial to identify conditions under which the extended Kalman filter provides accurate approximations. This paper generalizes two significant results from Picard (1991) on the efficiency of the continuous-time extended Kalman filter to a more general setting where the observation noise may be state-dependent but does not allow signal reconstruction from the quadratic variation of the observation process as in epidemic models. Firstly, we show that when the observation's drift coefficient is strongly injective and the signal's and observation's drift become nearly linear for the diffusion scaling coefficient $ε\to 0$, the estimation error is of order $\sqrtε$. Subsequently, we establish conditions under which the impact of the initial filtering error decays exponentially fast.

Figures (1)

• Figure 1: Flowchart of the $SI^{\pm}$ compartmental model.

Theorems & Definitions (11)

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