Chapter 3. Modeling and Simulation of Stochastic Processes


G. Q. Cai1, R. H. Huan2 and W. Q. Zhu2
Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, Florida, USA
2Department of Mechanics, State Key Laboratory of Fluid Power and Mechatronic and Control, Zhejiang University, Hangzhou, Zhejiang, China

Part of the book: Stochastic Processes: Fundamentals and Emerging Applications


For the purposes of analysis, numerical calculation and/or simulation, stochastic processes and their samples are required to be properly modeled and generated mathematically. Various methods have been developed to model stochastic processes according to their probabilistic and statistical properties. The present chapter is a review of these methods mostly based on the authors’ work in recent years. For a stochastic process, the two most important properties are its probability density function and power spectral density function. For a Gaussian process, four methods have been developed, the method of linear filters, the series expansions with random amplitudes, random phases, and both random amplitudes and random phases. To generate a non-Gaussian process defined in infinite range, semi-infinite range, or finite interval, two models are proposed, the nonlinear filter model and the randomized harmonic model. In some problems, two or more stochastic processes are involved, and they are correlated. Thus, feasible methods to model and generate correlated processes are needed. Two correlated stationary Gaussian processes are generated using the three methods, the method of linear filters, the method of series expansion with random amplitudes, and the method of series expansion with random phases. All three methods intend to match the power spectral density for each process, but use information of different levels of correlation. In the chapter, the systematic procedures are presented for these methods, and their application scopes and relative advantages are pointed out. Illustration examples are given to show the feasibility and accuracy of these methods.

Keywords: stochastic processes, Gaussian and non-Gaussian distributions, correlated processes, modeling, simulation


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