New Scheme for Estimation of the First and Senior Moment Functions for the Response of Linear Delay Differential System Excited by Additive and Multiplicative Noises

Abstract

The paper describes the theoretical apparatus and the algorithmic part of a scheme for stochastic examination of systems of linear stochastic delay differential equations perturbed by additive and multiplicative white noises. The presence of an efficient method of analyzing such problems, i.e., a method that can compete with the method of statistical simulation (Monte Carlo), is important both from the theoretical and applied points of view. The systems are substantial for modeling various objects and processes and can be the results of linearization of nonlinear stochastic systems with delay and multiplicative fluctuations. The importance of developing a procedure for calculating just the senior moment functions for the state vectors comes from the presence of multiplicative perturbations implying the inability to restrict the first moment functions as in the case of linear systems with Gaussian white noises as the input. To obtain ODEs for the first and senior moment functions without delays, we apply a new modification of our scheme combining the classical method of steps and extension of the system state space with the addition of the use of multidimensional matrices for the compact recording of the algorithm. The scheme realized in the environment of Wolfram Mathematica software package is demonstrated by an example devoted to an estimation the moment functions until the 4th order for a non-stationary stochastic response of the system with one degree of freedom.