This project is one of the projects of the National Natural Science Foundation of China. In the last twenty years, there’re many works carried out for dynamical problem of strong nonlinear systems subjected to Poisson white noise, at home and abroad. But little work has been done on research of random response prediction of hysteretic systems under excitations of Poisson white noise and its filtered processes. As new intelligent materials and energy dissipating elements with more complex properties of hysteresis have been brought into use, it is necessary to study the problem of random response prediction of these complex hysteretic systems under excitations of Poisson white noise and its filtered processes in order to insure that theoretical research can meet the engineering demands. In the present project, statistics of stationary response and transient response of Duhem hysteretic system and Preisach hysteretic system under excitations of Poisson white noise and linear/nonlinear filtered Poisson processes are studied to find out the effects of parameters of filtering systems and those of hysteretic systems on statistics of random response. The procedures of research are as follows. Firstly, the equivalent nonlinear system without hysteresis is generated from the simultaneous equations of filtering system and hysteretic system and the corresponding averaged generalized Fokker-Planck-Kolmogorov equation can be derived by using the stochastic averaging method. Then, an approximate solution is presented by using the exponential polynomial closure method to solve the averaged equation. Finally, by substituting the approximate solution into the Kolmogorov-Feller integro-differential equation to calculate the actual value of residual and modifying the approximate solution, a new approximate solution of statistics of response which meets the demand of error is obtained.