Stochastic filters attempt to estimate an unobservable state of a stochastic dynamical system from a set of noisy measurements. In this paper, we consider circular stochastic filtering and develop two dynamic methods for estimation of circular states, named sample-based stochastic filtering via root-finding (SB-SFRF) and Fourier-based stochastic filtering via root-finding (FB-SFRF). The proposed SB-SFRF and FB-SFRF methods attempt to dynamically minimize Bayes periodic risks by using Fourier series representation of their corresponding cost functions. The performance of the proposed methods is evaluated in the problem of direction-of-arrival (DOA) tracking.