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

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 of Picard (1991) on the efficiency of the continuous-time extended Kalman filter for a filtering system with small noise, 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 for example in epidemic models. First, we show that if the drift of the signal process and the observation process becomes nearly linear when the parameter $\epsilon$, which scales the diffusion coefficients, approaches zero, and the drift coefficient of the observation process is strongly injective, then the estimation error is of the order of $\sqrt{\epsilon}$. We then establish conditions under which the impact of the initial filtering error decays exponentially fast.