【作者】 牛原玲；

【导师】 张诚坚；

【作者基本信息】 华中科技大学 ， 概率论与数理统计， 2011， 博士

【摘要】 由于随机现象在自然界及其工程系统中的广泛存在,随机模型在生物、力学、经济、医学、工程等诸多科学领域中发挥着越来越重要的作用。同时,随着研究的不断深入,人们发现很多现象的发生会受到时滞因素的影响,即与事物的过去状态有关,而不是仅仅取决于系统的当前状态。随机泛函微分方程(SFDEs)常常被用来对这些系统进行建模,它们通常可以看成是泛函微分方程(FDEs)和随机微分方程(SDEs)的推广。由于随机泛函微分方程的显式解通常很难被求得,于是开展对其数值方法的研究就显得尤为重要。针对几类Ito型随机泛函微分方程,本文构造了若干新型数值方法,研究了其稳定性、收敛性及计算实现。特别地,我们探讨了生化系统的多尺度方法。全文组织如下：在第二章,我们给出了一种求解Ito型随机延迟微分方程的强预校方法,证明了在Lipschitz条件和线性增长条件下,该方法是min(1/2,p)阶收敛的。这里,被求解方程的初始函数是p次Holder连续的。其次,得到了该类方法的一个稳定性判据,结果表明：若对方法本身的自由参数p进行适当的选取,所得到的新方法将会比普遍使用的Euler-Maruyama方法具有更好的稳定性。数值结果验证了方法的收敛性,并且通过对其自由参数p的不同取值,对比了方法的稳定性。最后,讨论了方法的向量化实现,表明通过向量化实现可以使得方法的计算效率得以明显的提高。在第三章中,我们考虑了强预校方法应用于Ito型随机微分方程时的几乎必然指数稳定性和矩指数稳定性,得到了相应的稳定性判据。结果表明在一定的条件下,如果步长足够小,该方法是几乎必然指数稳定和矩指数稳定的。数值试验进一步验证了上述理论结果。在第四章,我们提出了一种求解随机延迟微分方程的显式强1阶无导数方法,其中,所研究的随机延迟微分方程要求具有足够光滑的漂移系数和扩散系数以及标量型的维纳过程。此外,我们也给出了一个Mil stein方法求解线性测试方程的稳定性结论,由该结论得到的稳定域较先前文献给出的稳定域要更大。为了对比方法的稳定域,我们进一步研究了无导数方法和Milstein方法求解线性随机延迟微分方程的稳定性举止。最后,用数值试验证实了其稳定性结论。在第五章考虑了一类带随机扰动和记忆项的复杂系统,这类系统可以用非线性随机延迟积分微分方程来进行建模。本章中,我们得到了一个关于该方程的延迟依赖的稳定性判据,并且利用数值试验进一步论证了上述理论结果。在第六章,我们给出了分子数目跨度很大的延迟生化系统的多尺度模拟方法。基于系统分割的思想和已有的延迟生化系统的仿真方法,提出了一种可以显著降低计算复杂度的新方法。结果表明,与已有的仿真方法(如DSSA方法和改进的next reaction方法)比较,多尺度方法具有更高的效率。最后,用数值试验验证了方法的精确性和高效性。更多还原

【Abstract】 Random fluctuations are abundant in natural or engineered systems. Therefore, stochastic modelling has come to play an important role in various fields like biology, me-chanics, economics, medicine and engineering. Moreover, these systems are sometimes sub-ject to memory effects, when their time evolution depends on their past history with noise disturbance. Stochastic functional differential equations (SFDEs) are often used to model such systems. They can be regarded as a generalization of both deterministic functional differential equations (FDEs) and stochastic ordinary delay differential equations (SODEs). Explicit solutions of SFDEs can rarely be obtained. Thus, it has become an important issue to develop numerical methods for SFDEs.This thesis is devoted to the investigation of numerical algorithms for several classes of stochastic functional differential equations. The stability, convergence and computational implementation of these methods are analyzed. In particular, a multi-scale approach that can reduce the computational burden is constructed on the basis of the predictor-corrector method given previously. The construction of the thesis is as follows.Chapter 2 presents a strong predictor-corrector method for the numerical solution of stochastic delay differential equations (SDDEs) of Ito-type. The method is proved to be mean-square convergent of order min(1/2.p) under the Lipschitz condition and the linear growth condition, where p is the exponent of Holder condition of the initial function. A stability criterion for this type of method is derived. It is shown that for certain choices of the flexible parameter p the derived method can have a better stability property than more commonly used numerical methods. Numerical results are reported confirming con-vergence properties and comparing stability properties of methods with different parameters p. Finally, the vectorised simulation is discussed and it is shown that this implementation is much more efficient. Chapter 3 deals with almost sure and moment exponential stability of a class of predictor-corrector methods applied to the stochastic differential equations of Ito-type. Sta-bility criteria for this type of methods are derived. The methods are shown to maintain almost sure and moment exponential stability for all sufficiently small timesteps under ap-propriate conditions. Numerical experiment further testifies these theoretical results.In chapter 5, a new explicit stochastic scheme of strong order 1 is proposed for stochas-tic delay differential equations (SDDEs) with sufficiently smooth drift and diffusion coeffi-cients and a scalar Wiener process. The method is derivative-free and is shown to be stable in mean square. A stability theorem for the continuous strong approximation of the solution of a linear test equation by the Milstein method is also proved, which shows the stability bound is larger than bounds given previously in the literature. The case of linear SDDEs is further investigated, in order to compare the stability bounds of these two methods. Numer-ical experiments are given to confirm the stability properties obtained.Chapter 6 presents a multi-scale approach for simulating time-delay biochemical reaction systems when there are wide ranges of molecular numbers. We construct a new approach that can reduce the computational burden on the basis of the idea of a partitioned system and recent developments with stochastic simulation algorithm and the delay stochastic simulation method. It is shown that this algorithm is much more efficient than existing methods such as DSSA method and the modified next reaction method. Some numerical results arc reported, confirming the accuracy and computational efficiency of the approximation.更多还原