Modelling of high-dimensional diffusion stochastic process with nonlinear coefficients for engineering applications - part II: Approximations for covariance and spectral density of stationary process

This work is devoted to diffusion stochastic processes (DSPs) with nonlinea r coefficients in n-dimensional Euclidean space at high n (n is much greate r than a few units). It deals with covariance and spectral density of a sta tionary process whereas our accompanying work deals with expectation and va riance of a nonstationary process. Combined, analytical-numerical approach is a reasonable and perhaps the only way to treat high-dimensional DSPs in practice. Each of the above works develops the corresponding parts of the a nalytical basis for this combined treatment. The present work proposes appr oximate 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 de rived within DSP theory. No techniques directly related to stochastic diffe rential equations are involved. The approximate expression for the covarian ce is also applied to evaluate the approximation for the spectral density. Some aspects of practical implementation of these approximations are discus sed. The results of this work can equally be used in various engineering fi elds.