Stochastic modeling of nonlinear oscillators under combined Gaussian and Poisson white noise: a viewpoint based on the energy conservation law

Abstract

A stochastic differential equation model is considered for nonlinear oscillators under excitations of combined Gaussian and Poisson white noise. Since the solutions of stochastic differential equations can be interpreted in terms of several types of stochastic integrals, it is sometimes confusing about which integral is actually appropriate. In order for the energy conservation law to hold under combined Gaussian and Poisson white noise excitations, an appropriate stochastic integral is introduced in this paper. This stochastic integral reduces to the Di Paola–Falsone integral when the multiplicative noise intensity is infinitely differentiable with respect to the state. The stochastic integral introduced in this paper is applicable in more general situations. Numerical examples are presented to illustrate the theoretical conclusion.