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

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.