The scheme for calculating the covariance functions of the state vectors for non-stationary linear stochastic differential systems with delay

The paper describes the method and algorithm of analysis in the time domain of linear systems of stochastic ordinary differential equations (SODE) with variable coefficients and constant delays, designed to obtain the covariance functions of state vectors, in the case of assumption that stochastic perturbations are Gaussian non-stationary random noises. To solve the problem, we propose a multi-step scheme. Each of steps consists of three stages. The first based on the classical method of steps, expands the state space which ultimately allows us to obtain a chain of linear SODE systems without delay; on the second step, using the correlation theory, systems of linear ODE are constructed for vector functions of expectation and matrix covariance functions with the necessary initial conditions; at the third stage, we apply the correlation theory too and the results of the second stage and achieve the required result of the current step.