随机延迟微分方程分裂步θ方法的数值分析

发布时间：2018-04-22 06:41

  本文选题：随机延迟微分方程 + 扩散的分裂步θ方法　； 参考：《哈尔滨工业大学》2017年硕士论文

【摘要】：随机延迟微分方程作为一种重要的数学模型在物理学,生物学,金融学,控制论以及医学等诸多领域具有广泛的应用。这一类方程既考虑了滞后对系统的作用,同时考虑了外界环境对系统性质所造成的影响。因此,随机延迟微分方程更加准确的模拟了自然生活。在实际应用中,随机延迟微分方程精确解的显式表达式很难求出,或表达式很复杂,因此构造适用的数值方法并研究数值方法的性质具有重要意义。近年来许多学者研究了随机常延迟微分方程及数值方法,对于随机变延迟微分方程及数值方法的研究刚刚开始。本文探讨了随机变延迟微分方程分裂步θ方法的收敛性和稳定性。本文分别研究了扩散的分裂步θ方法和漂移的分裂步θ方法的收敛性和稳定性。首先,当随机延迟微分方程的系数满足全局Lipschitz条件和线性增长条件时,研究了扩散的分裂步θ方法的均方收敛性,并得出该方法的均方收敛阶为1/2;随后本文在单调条件和线性增长条件下探讨了扩散的分裂步θ方法的稳定性,证明了该方法对于一定范围内的步长是均方指数稳定的。其次,当随机延迟微分方程的系数满足全局Lipschitz条件和线性增长条件时,分析了漂移的分裂步θ方法的均方收敛性,并得出该方法的均方收敛阶为1/2;同时在单调条件和线性增长条件下,本文探讨了漂移的分裂步θ方法的稳定性,证明了当θ∈(1/2,1]时,该方法依任意步长保持均方指数稳定;当θ∈[0,1/2]时,存在h0,使得当h∈(0,h0)时,漂移的分裂步θ方法是均方指数稳定的。
[Abstract]:As an important mathematical model, stochastic delay differential equations are widely used in physics, biology, finance, cybernetics and medicine. This kind of equation not only considers the effect of hysteresis on the system, but also considers the influence of the external environment on the properties of the system. Therefore, the stochastic delay differential equations more accurately simulate the natural life. In practical application, the explicit expression of exact solution of stochastic delay differential equation is very difficult to obtain, or the expression is very complicated, so it is very important to construct the suitable numerical method and study the properties of the numerical method. In recent years, many scholars have studied stochastic constant delay differential equations and numerical methods, and the research on stochastic variable delay differential equations and numerical methods has just begun. In this paper, the convergence and stability of the splitting step 胃 method for stochastic variable delay differential equations are discussed. In this paper, the convergence and stability of diffusion splitting 胃 method and drift splitting step 胃 method are studied respectively. Firstly, when the coefficients of the stochastic delay differential equation satisfy the global Lipschitz condition and the linear growth condition, the mean square convergence of the split step 胃 method of diffusion is studied. The mean square convergence order of the method is 1 / 2. Then the stability of the diffusion splitting step 胃 method is discussed under the monotone condition and the linear growth condition. It is proved that the method is exponentially stable for the step size in a certain range. Secondly, when the coefficients of the stochastic delay differential equation satisfy the global Lipschitz condition and the linear growth condition, the mean square convergence of the drift splitting step 胃 method is analyzed. At the same time, under monotone condition and linear growth condition, the stability of the drift splitting step 胃 method is discussed, and it is proved that the mean square exponent stability of the method is maintained according to arbitrary step size when 胃 鈭，

本文编号：1786084