Numerical Analysis Of A Linear Stochastic Oscillator With Additive Noise

Abstract:Stochastic differential equation is not only widely used in natural science, such as mathematical science, physics and so on, but also is one of the most commonly used mathematical models in engineering, economic management and financial engineering.This paper studies one of the second order stochastic differential equation models-the linear stochastic oscillator with additive noise, and discusses some numerical methods’ability in preserving the linear growth rate of the energy of this stochastic system. The numerical methods used in this paper including the Euler-Maruyama method (EM), the Backward Euler Method (BE), the partitioned Euler-Maruyama method (PEM) and the implicit midpoint method (IM). At the same time we also study the predictor-correct methods (P(EC)k) formed by the above four methods. The main contents of this paper include:In the first chapter, the background, numerical simulation and present research situation of the linear stochastic oscillator are reviewed.In the second chapter, some related preliminary knowledge is outlined, mainly including some basic knowledge of stochastic analysis, the basic concepts of stochastic differential equation, as well as its numerical simulation. In the third chapter, we discuss the EM, BE, PEM and midpoint method in preserving the linear growth of the energy of linear stochastic oscillator with additive noise. For low frequency, the three methods’ simulation effect is well. The theoretical results show that the EM, BE and PEM have the same weak order of convergence1, and the IM method has weak order of convergence2.In the fourth chapter, some predictor-correct methods consisted by EM, BE, PEM and IM are studied. We can proof that predictor-correct methods have weak order of convergence2, we also study their ability of preserving the area of their phase flow.In, the fifth chapter, numerical experiments to verify the fourth chapter are given. The results show that although predictor-correct methods can reduce the error, they applicable only to low frequency of the stochastic oscillator. When BE is the predict operator, IM is the correct operator, the scope of the frequency has been somewhat expanded, compared with EM for the predict operator, PEM for the correct operator.