Modelling of High-Dimensional Diffusion Stochastic Process With Nonlinear Coefficients for Engineering Applications — Part Ii: Approximations for Covariance and Spectral Density of Stationary Process

Abstract

This work is devoted to diffusion stochastic processes (DSPs) with nonlinear coefficients in n-dimensional Euclidean space at high n (n is much greater than a few units). It deals with covariance and spectral density of a stationary process whereas our accompanying work deals with expectation and variance of a nonstationary process. Combined, analytical-numerical approach is a reasonable and perhaps the only way to treat highdimensional DSPs in practice. Each of the above works develops the corresponding parts of the analytical basis for this combined treatment. The present work proposes approximate analytical expressions for the covariance in the efficient integral form. The integral formula is based on the obtained analytical results for the covariance, allows for nonlinearities of the drift function, and is derived within DSP theory. No techniques directly related to stochastic differential equations are involved. The approximate expression for the covariance is also applied to evaluate the approximation for the spectral density. Some aspects of practical implementation of these approximations are discussed. The results of this work can equally be used in various engineering fields.