Heuristic method of adaptive filtering for noisy GARCH processes

Abstract

In this paper, the filtering problems of the discrete systems with noisy GARCH(1,1) processes are discussed by the heuristic method. Based on the intuition that the density function of the noisy GARCH(1,1) process can be well approximated by its driving noise density function with time-varying variance, three different approximating methods are proposed, respectively. The first is to approximate the density function of the system measurement noise by its driving noise density function with time-varying variance, then calculate the iterative state vector and measurement vector. The second is only to use its driving noise density function to approximate the density function of the state noisy GARCH(1,1) process, and then calculate the iterative state and measurement vector. The third is to use its driving noise density functions with time-varying variances to directly approximate the density functions of the noisy GARCH(1,1) processes for the state and measurement vectors. The numerical simulation results demonstrate the effectiveness of our algorithms in the noisy GARCH(1,1) environment.