On Bounded Stochastic Processes

Abstract

Stochastic processes of bounded variation are generated based on their two most important characteristics: spectral density functions and probability density functions. Two models are presented for the purpose: the randomized harmonic model and the nonlinear filter model. In the randomized harmonic model, a random noise is introduced in the phase angle; while in the nonlinear filter model, a set of nonlinear Ito differential equations are employed. In both methods, the spectral density of a stochastic process to be modeled, either with one peak or with multiple peaks, can be matched by adjusting model parameters. However, the probability density of the process generated by the randomized harmonic model has a fixed shape and cannot be adjusted. On the other hand, the nonlinear filter model covers a variety of profiles of probability distributions. For the Monte Carlo simulation using these two models, equivalent and alternative expressions are proposed, which make the simulation more effective and efficient.