A class of weakly stationary non-Gaussian models

Abstract

Two models are developed for real-valued, weakly stationary non-Gaussian processes using the spectral representation X ( t ) = ∫ 0 ∞ [ cos ( ν t ) d M 1 ( ν ) + sin ( ν t ) d M 2 ( ν ) ] . The processes M i , i = 1 , 2 , in this representation are (1) square integrable martingales defined as functions H n ( B i ( ν ) , ν ) , ν ≥ 0 , depending on Hermite polynomials of independent Brownian motions B i , i = 1 , 2 , or (2) have increments d M i ( ν ) = G i ( ν ) d B i ( ν ) , ν ≥ 0 , where G i are processes with finite variance independent of B i , i = 1 , 2 . It is shown that the models can be calibrated to any target spectral density. Efficient methods are developed for calculating their higher order statistics. Features and limitations of the proposed models are illustrated by numerical examples. Monte Carlo algorithms for generating non-Gaussian samples based on the models in the paper resemble classical spectral representation-based algorithms for generating stationary Gaussian samples.