Random scale perturbation of an AR(1) process and its properties as a nonlinear explicit filter

Abstract

We study the properties of a nonlinear model of filtering in discrete time which leads to explicit computations. The signal is a standard AR(1) process, but noises are multiplicative and non-Gaussian. If the initial distribution of the AR(1) process is taken to belong to a specified class, the prediction and optimal filters also belong to this class and the prediction and updating steps are explicit. We prove the existence of a stationary version for the prediction filter and complete the theoretical study with simulations to illustrate the behaviour of the filter.