Direct derivation of response moment and cumulant equations for non-linear stochastic problems

Abstract

A relatively straightforward formulation is presented for deriving the differential equations governing the evolution of the response moments and cumulants of a dynamical system. This is a very general framework that applies to linear and non-linear systems subjected to external and multiplicative non-Gaussian, delta-correlated processes. This formulation provides an alternative to both the partial differential Fokker-Planck equation that has sometimes been used in deriving moment or cumulant equations, and the differential (as opposed to derivative) relationships of the Ito calculus. It is believed that many analysts may find the technique used here to be more obvious than the alternatives, since the derivative relationships for the stochastic process are of the same form as in the more familiar ordinary differential equations, (C) 2000 Elsevier Science Ltd. All rights reserved.