几类随机泛函微分方程的数值算法与理论
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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。
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.
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。
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.
引文
[1]Xu D, Feng Z, Allen L J S, et al. A spatially structured metapopulation model with patch dynamics. J. Theor. Biol.,2006,239(4):469-481.
[2]Parsons D J, Benjamin L R, Clarke J, et al. Weed Manager—A model-based decision support system for weed management in arable crops. Comput. Electron. Agric.,2009,65(2):155-167.
[3]Mortimer D, Dayan P, Burrage K, et al. Optimizing chemotaxis by measuring unbound-bound transitions. Physica D:Nonlinear Phenom.,2009,239(9):477-484.
[4]Allen L J S, Flores D A, Ratnayake R K, et al. Discrete-time deterministic and stochastic models for the spread of rabies. Appl. Math. Comput.,2002,132(2-3):271-292.
[5]Golightly A, Wilkinson D J. Bayesian inference for nonlinear multivariate diffusion models ob-served with error. Comput. Statistics Data Anal.,2008,52:1674-1693.
[6]Martin C F, Allen L J S. Urn Model Simulations of a Sexually Transmitted Disease Epidemic. Appl. Math. Comput.,1995,71(2-3):179-199.
[7]Hofmann N, Platen E, Schweizer M. Option pricing under incompleteness and stochastic volatility. Math. Finance,1992,2(3):153-187.
[8]Harris C J. Poisson Runge-Kutta methods for chemical reaction systems. in:Brebbi C A, editor, Proceedings of Mathematical Models and Environmental Problems, London:Pentech Press,1976, 269-282.
[9]Oksendal B. Stochastic differential equations:An introduction with applications. Berlin:Springer-Verlag,2003.
[10]Carletti M. Stochastic modelling of biological processes:[PhD Dissertation]. Brisbane, Australia: The University of Queensland,2008.
[11]Hasty J, Isaacs F, Dolnik M, et al. Designer gene networks:Towards fundamental cellular control. Chaos,2001,11(1):207-220.
[12]Burrage K, Tian T. Poisson Runge-Kutta methods for chemical reaction systems. in:Lu Y, Sun W, Tang T, editors, Proceedings of Advances in Scientific Computing and Applications, Beijing/New York:Science Press,2004,82-96.
[13]Tomioka R, Kimura H, Kobayashi T J, et al. Multivariate analysis of noise in genetic regulatory networks. J. Theor. Biol.,2004,229(4):501-521.
[14]Gillespie D T. The chemical Langevin equation. J. Chem. Phys.,2000,113(1):297-306.
[15]Gillespie D T. A rigorous derivation of the chemical master equation. Physica A,1992,188:404-425.
[16]Melykuti B, Burrage K, Zygalakis K C. Fast stochastic simulation of biochemical reaction sys-tems by alternative formulations of the chemical Langevin equation. J. Chem. Phys.,2010, 132(16):164109.
[17]Tian T, Burrage K. Bistability and switching in the lysis/lysogeny genetic regulatory network of bacteriophage A. J. Theor. Biol.,2004,227(2):229-237.
[18]Turner T E, Schnell S, Burrage K. Stochastic approaches for modelling in vivo reactions. Comput. Biol. Chem.,2004,28(3):165-178.
[19]Golightly A, Wilkinson D J. Bayesian inference for nonlinear multivariate diffusion models ob-served with error. Comput. Statistics Data Anal.,2008,52(3):1674-1693.
[20]Kesinger J C, Allen L J. Genetic models for plant pathosystems. Math. Biosci.,2002,177-178:247-269.
[21]Marquez-Lago T T, Leier A, Burrage K. Probability distributed time delays:integrating spatial effects into temporal models. BMC Syst. Biol.,2010,4(19):1-16.
[22]Bernard S, Cajavec B, Pujo-Menjouet L, et al. Modeling transcriptional feedback loops:The role of Gro/TLE1 in Hesl oscillations. Phil Transact A Math Phys. Eng. Sci.,2006,364(1842):1155-1170.
[23]Lewis J. Autoinhibition with transcriptional delay:a simple mechanism for the zebrafish somito-genesis oscillator. Curr. Biol.,2003,13(16):1398-1408.
[24]Hirata H, Yoshiura S, Ohtsuka T, et al. Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop. Science,2002,298(5594):840-843.
[25]Carletti M. Numerical solution of stochastic differential problems in the biosciences. J. Comput. Appl. Math.,2006,185(2):422-440.
[26]Carletti M. Mean-square stability of a stochastic model for bacteriophage infection with time delays. Math. Biosci.,2007,210(2):395-414.
[27]Carletti M. Numerical simulation of a Campbell-like stochastic delay model for bacteriophage infection. Math. Med. Biol.,2006,23(4):297-310.
[28]Tian T, Burrage K, Burrage P M, et al. Stochastic delay differential equations for genetic regulatory networks. J. Comput. Appl. Math.,2007,205(2):696-707.
[29]Mao X. Stochastic differential equations and their applications. Chichester:Horwood,1997.
[30]胡适耕,黄乘明,吴付科.随机微分方程.北京：科学出版社,2008.
[31]Ito K. Introduction to Probability Theory. Cambridge:Cambridge University Press,1984.
[32]Kunita H. Stochastic flows and stochastic differential equations. Cambridge:Cambridge Univer-sity Press,1990.
[33]Malliavin P. Stochastic analysis. Berlin:Springer,1997.
[34]Arnold L. Stochastic differential equations:theory and applications. New York:Wiley,1974.
[35]Sobcayk K. Stochastic differential equations, with applications to Physics and Engineering. Dor-drecht:Kluwer Academic Publishers,1991.
[36]Skorokhod A V. Studies in the theory of random processes. Reading, Mass:Addison-Wesley, 1965.
[37]Kloeden P E, Platen E. Numerical solution of stochastic differential equations. Berlin:Springer-Verlag,1992.
[38]Gard T C. Introduction to stochastic differential equations. New York:Marcel Dekker,1988.
[39]Platen E. An introduction to numerical methods for stochastic differential equations. Acta Numer., 1999,8:197-246.
[40]Milstein G N. Approximate integration of stochastic differential equations. Theory Prob. Appl., 1974,19(3):557-562.
[41]Platen E. Zur zeitdiskreten approximation von itoprozessen. Diss. B., Imath, Akad, der DDR, Berlin,1984.
[42]W.Rumelin. Numerical treatment of stochastic differential equations. SIAM J. Numer. Anal., 1982,19(3):604.
[43]Burrage K, Burrage P M. Order conditions of stochastic Runge-Kutta methods by B-series. SIAM J. Numer. Anal.,2000,38(5):1626-1646.
[44]王志勇.随机泛函微分方程的稳态数值解研究：[博士学位论文].武汉：华中科技大学,2008.
[45]Tian T, Burrage K. Implicit Taylor methods for stiff stochastic differential equations. Appl. Numer. Math.,2001,38(1-2):167-185.
[46]Burrage K, Tian T. Stiffly accurate Runge-Kutta methods for stiff stochastic differential equa-tions. Comput. Phys. Commun.,2001,142:186-190.
[47]Burrage K, Tian T. Implicit stochastic Runge-Kutta methods for stochastic differential equations. BIT Numer. Math.,2004,44(1):21-39.
[48]Debrabant K, Roβer A. Continuous weak approximation for stochastic differential equations. J. Comput. Appl. Math.,2008,214(1):259-273.
[49]Talay D. Second-order discretization schemes of stochastic differential systems for the computa-tion of the invariant law. Stochastics Stochastic Rep.,1990,29(1):13-36.
[50]Roβer A. Rooted tree analysis for order conditions of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations. Stochastic Anal. Appl.,2006,24(1):97-134.
[51]Komori Y. Weak first-or second-order implicit Runge-Kutta methods for stochastic differential equations with a scalar Wiener process. J. Comput. Appl. Math.,2008,217(1):166-179.
[52]Saito Y, Mitsui T. Stability analysis of numerical schemes for stochastic differential equations. SLAM J. Numer. Anal.,1996,33(6):2254-2267.
[53]Saito S, Mitsui T. T-stability of numerical scheme for stochastic differential equations. World Sci. Ser. Anal.,1993,2:333-344.
[54]Burrage K, Tian T. A note on the stability properties of the Euler methods for solving stochastic differential equations. New Zealand J. of Maths,2000,29:115-127.
[55]D.J.Higham, Mao X, Yuan C. Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. SAIM J. Numer. Anal.,2007,45(2):592-609.
[56]Pang S, Deng F, Mao X. Almost sure and moment exponential stability of Euler-Maruyama discretizations for hybrid stochastic differential equations. J. Comput. Appl. Math.,2008, 213(1):127-141.
[57]Hofmann N. Stability of weak numerical shemes for stochastic differential equations. Math. Comput. Simul.,1995,38(1-2):63-68.
[58]Hofmann N, Platen E. Stability of weak numerical shemes for stochastic differential equations. Comput. Math. Appl.,1994,28:45-47.
[59]Mohammed S. Stochastic functional differential equations. New York:Longman Scientific and Technical,1986.
[60]Ivanov A, Kazmerchuk Y, Swishchuk A. Theory, stochastic stability and applications of stochastic differential equations:a survey of recent results. Diff. Equa. Dyn. Syst.,2003,11(1-2):55-115.
[61]Longtin A. Stochastic delay-differential equations. in:Proceedings of Complex Time-delay Sys-tems. Springer Berlin,2009,177-195.
[62]Buckwar E. Introduction to the numerical analysis of stochastic delay differential equations. J. Comput. Appl. Math.,2000,125(1-2):297-307.
[63]Baker C T H, Buckwar E. Numerical analysis of explicit one-step methods for stochastic delay differential equations. LMS J. Comput. Math.,2000,3:315-335.
[64]Kuchler U, Platen E. Stong discrete time approximation of stochastic differential equations with time delay. Math. Comput. Simulat.,2000,54(1):189-205.
[65]Liu M, Cao W, Fan W. Convergence and stability of the semi-impplicit Euler method for a linear stochastic differential delay equation. J. Comput. Appl. Math.,2004,170(2):255-268.
[66]Buckwar E. One-step approximations for stochastic functional differential equations. Appl. Nu-mer. Math.,2006,56(5):667-681.
[67]覃婷婷.几类随机与延迟动力学系统的单步离散方法：[博士学位论文].武汉：华中科技大学,2010.
[68]Hu L, Gan S. Convergence and stability of the balanced methods for stochastic differential equa-tions with jumps. Int. J. Comput. Math.,2011,99999(1):1-20.
[69]Cao W, Liu M, Fan Z. MS-stability of the Euler-Maruyama method for stochastic differential delay equations. Appl. Math. Comput.,2004,159(1):127-135.
[70]Baker C T H, Buckwar E. Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations. J. Comput. Appl. Math.,2005, 184(2):404-427.
[71]曹婉容,刘明珠.随机延迟微分方程Euler-Maruyama数值方法的T-稳定性.哈尔滨工业大学学报,2005,37(3)：303-305.
[72]Wang Z, Zhang C. An analysis of stability of Milstein method for stochastic differential equations with delay. Comput. Math. Appl.,2006,51(9-10):1445-1452.
[73]王文强,黄山,李寿佛.非线性随机延迟微分方程Euler-Maruyama方法的均方稳定性.计算数学,2007,29(2)：217-224.
[74]王文强.几类非线性随机延迟微分方程数值方法的收敛性与稳定性：[博士学位论文].湘潭：湘潭大学,2007.
[75]王文强,黄山,李寿佛.非线性随机延迟微分方程半隐式Euler方法的收敛性.数值计算与计算机应用,2008,29(1)：73-80.
[76]王志勇,张诚坚.随机延迟微分方程的Milstein方法的非线性均方稳定性.应用数学,2008,21(1)：201-206.
[77]范振成.随机延迟微分方程Mil stein方法的均方稳定性.应用数学,2008,21(4)：743-747.
[78]Zhang H, Gan S, Hu L. The split-step backward Euler method for linear stochastic differential equations. J. Comput. Appl. Math.,2009,225(2):558-568.
[79]Wang X, Gan S. The improved split-step backward Euler method for stochastic differential delay equations. to be in Int. J. of Comput. Math.
[80]范振成.随机延迟微分方程的全隐式Euler方法.计算数学,2009,31(3)：287-298.
[81]王文强.非线性随机延迟微分方程Milstein方法的均方稳定性.系统仿真学报,2009,21(18)：5656-5658.
[82]Baker C, Buckwar E. Continuous θ-methods for the pantograph equation. Electron. Trans. Numer. Anal.,2000,11:131-151.
[83]Fan Z, Liu M, Cao W. Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations. J. Math. Anal. Appl.,2007,325(2):1142-1159.
[84]范振成,刘明珠.线性随机比例方程的渐近均方稳定性.应用数学,2007,20(3)：519-523.
[85]肖飞雁.几类随机延迟微分代数系统的数值分析：[博士学位论文].武汉：华中科技大学,2008.
[86]Golec J, Sathananthan S. Sample path approximation for stochastic integro-differential equations. Stoch. Anal. Appl.,1999,17(4):579-588.
[87]Golec J, Sathananthan S. Strong approximations of stochastic integro-differential equations. Dyn. Contin. Discrete Implus. Syst. Ser. B Appl. Algorithms,2001,8:139-151.
[88]Buckwar E. The theta-Maruyama scheme for stochastic functional differential equations with distributed memory term. Monte Carlo Methods Appl.,2004,10:235-244.
[89]Shaikhet L, Roberts J. Reliability of difference analogues to preserve stability properties of stochastic Volterra integro-differential equations. Adv. Difference Equ.,2006,2006(1):1-22.
[90]Ding X, Wu K, Liu M. Convergence and stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations. Int. J. Comput. Math.,2006,83(10):753-763.
[91]Rathinasamy A, Balachandran K. Mean-square stability of Milstein method for linear hybrid stochastic delay integro-differential equations. Nonlinear Anal. Hybrid Syst.,2008,2(4):1256-1263.
[92]Hairer E, Wanner G. Solving ordinary differential equations II:Stiff and differential-algebraic Problems. Berlin:Springer-Verlag,1996.
[93]Kloeden P E, Platen E. Higher-order implicit strong numerical scheme for stochastic differential equations. J. Stat. Phys.,2001,66(1):283-314.
[94]Milstein G N, Platen E, Schurz H. Balanced implicit methods for stiff stochastic systems. SIAM J. Numer. Anal.,1998,35(3):1010-1019.
[95]Alcock J, Burrage K. A note on the balanced method. BIT,2006,46(4):689-710.
[96]Brugnano L, K.Burrage, M.Burrage P. Adams-Type methods for the numerical solution of stochas-tic ordinary differential equations. BIT,2000,40(3):451-470.
[97]Burrage K, Tian T. Predictor-corrector methods of Runge-Kutta type for stochastic differential equations. SIAM J. Numer. Anal.,2002,40(4):1516-1537.
[98]Bruti-Liberati N, Platen E. Strong predictor-corrector Euler methods for stochastic differential equations. Stoch. Dynam.,2008,8(3):561-581.
[99]Elowitz M B, Leibler S. A synthetic oscillatory network of transcriptional regulators. Nature, 2000,403(6767):335-338.
[100]Sano J. Random monoallelic expression of three genes clustered within 60 kb of mouse t complex genomic DNA. Genome Res.,2001,11(11):1833-1841.
[101]Federoff N, Fontana W. Small numbers of big molecules. Science,2002,297(5584):1129-1131.
[102]Hume D A. Probability in transcriptional regulation and its implications for leukocyte differentia-tion and inducible gene expression. Blood,2000,96(7):2323-2328.
[103]Kepler T, Elston T. Stochasticity in transcriptional regulation:origins, consequences, and mathe-matical representations. Biophys. J.,2001,81 (6):3116-3136.
[104]McAdams H H, Arkin A. It's a noisy business! Genetic regulation at the nanomolar scale. Trends Genet.,1999,15(2):65-69.
[105]Anderson D F. A modified next reaction method for simulating chemical systems with time de-pendent propensities and delays. J. Chem. Phys.,2007,127(21):4107.
[106]Gibson M A, Bruck J. Efficient exact stochastic simulation of chemical systems with many species and many channels. J. Phys. Chem.,2000,104(9):1876-1889.
[107]Gillespie D T. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem.,1977, 81(25):2340-2361.
[108]Gillespie D T. Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys.,2001,115(4):1716-1733.
[109]Tian T, Burrage K. Binomial leap methods for simulating stochastic chemical kinetics. J. Chem. Phys.,2004,121 (21):10356-10364.
[110]Jensen M H, Sneppen K, Tiana G. Sustained oscillations and time delays in gene expression of protein Hesl. FEBS Lett.,2003,541 (1):176-177.
[111]Monk N A M. Oscillatory expression of Hesl, p53, and NF-jB driven by transcriptional time delays. Curr. Biol.,2003,13(16):1409-1413.
[112]Beretta E, Carletti M, Solimano F. On the effects of environmental fluctuations in a simple model of bacteria-bacteriophage interaction. Canad. Appl. Math. Quart.,2000,8(4):321-366.
[113]Cooke K, Driessche P, Zou X. Interaction of maturation delay and nonlinear birth in population and epidemic model. J. Math. Biol.,1999,39(4):332-352.
[114]Liu S, Beretta E. A stage-structured predator-prey model of Beddington-Deangelis type. SIAM J. Appl. Math.,2006,66(4):1101-1129.
[115]Chen L, Wang R, Zhou T. Noise-induced cooperative behavior in a multi-cell system. Bioinfor-matics,2005,21(11):2722-2729.
[116]Barrio M, K B, Leier A, et al. Oscillatory regulation of Hesl:discrete stochastic delay modelling and simulation. PLoS Comp. Biol.,2006,2(9):1017-1030.
[117]Bratsun D, Volfson D, Tsimring L S, et al. Delay-induced stochastic oscillations in gene regulation. in:Proceedings of Proc. Natl. Acad. Sci. U.S.A. National Academy of Sciences,2005,14593-14598.
[118]Cai X. Exact stochastic simulation of coupled chemical reactions with delays. J. Chem. Phys., 2007,126(12):4108.
[119]Haseltine E L, Rawlings J B. Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys.,2002,117(15):6959-6969.
[120]Rao C, Arkin A. Stochastic chemical kinetics and the quasi-steady-state assumption:application to the Gillespie algorithm. J. Chem. Phys.,2003,118(11):4999-5010.
[121]Burrage P M. Vectorised simulations for stochastic differential equations. ANZIAM J.,2004, 45(E):C350-C363.
[122]Kloeden P E, Platen E. Numerical solution of stochastic differential equations. Berlin:Springer, 1995.
[123]Burrage K, Burrage P M. High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations. Appl. Numer. Math.,1996,22(1-3):81-101.
[124]Burrage P M. Runge-Kutta methods for stochastic differential equations:[PhD Dissertation]. Queensland, Australia:The University of Queensland,1999.
[125]J.Higham D. An algorithmic introduction to numerical simulation of stochastic differential equa-tions. SIAM Review,2001,43(3):525-546.
[126]Milstein G N. Numerical integration of stochastic differential equations. Dordrecht:Kluwer Academic Publisher,1995.
[127]J.Higham D. Mean-square and asymptotic stability of numerical methods for stochastic ordinary differential equations. SIAM J. Numer. Anal.,2000,38(3):753-769.
[128]Buckwar E, Winkler R. Multistep methods for SDEs and their application to problems with small noise. SAIM J. Numer. Anal.,2006,44(2):779-803.
[129]Carletti M, Burrage K, Burrage P M. Numerical simulation of stochastic ordinary differential equations in biomathematical modelling. Math. Comput. Simul.,2004,64(2):271-277.
[130]Xu Y, Allena L J S, Perelson A S. Stochastic model of an influenza epidemic with drug resistance. J. Theor. Biol.,2007,248(1):179-193.
[131]Cao Y, Samuels D. Discrete stochastic simulation methods for chemically reacting systems. Meth-ods Enzymol.,2009,454:115-140.
[132]Higham D J. Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal.,2000,38(3):753-769.
[133]Bryden A, Higham D J. On the boundedness of asymptotic stability regions for the stochastic theta method. BIT Numer. Math.,2003,43(1):1-6.
[134]Higham D J, Mao X, Stuart A M. Exponential mean-square stability of numerical solutions to stochastic differential equations. LMS J. Comput. Math.,2003,6:297-313.
[135]Niu Y, Zhang C, Burrage K. Strong predictor-corrector approximations for SDDEs of Ito-type. submitted.
[136]Mao X, Sabanis S. Numerical solutions of stochastic differential delay equations under local Lipschitz conditions. J. Comp. Appl. Math.,2003,151(1):215-227.
[137]Hu Y, Mohammed S E A, Yan F. Discrete-time approximations of stochastic delay equations:the Milstein scheme. Ann. Probab.,2004,32(1):265-314.
[138]Lei J, Mackey M. Stochastic differential delay equation, moment stability, and application to hematopoietic stem cell regulation system. SIAM J. Appl. Math.,2007,67(2):387-407.
[139]Ortalo-Magne F, Rady S. Heterogeneity within communities:A stochastic model with tenure choice. J. Urban Economics,2008,64(1):1-17.
[140]Tass P. Phase resetting in medicine and biology:Stochastic modelling and data analysis. Berlin: Springer Verlag,1999.
[141]Thomson D J. A stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence, and its application to the problem of concentration variance. J. Fluid Mech., 1990,210:113-153.
[142]Appleby J A D, Riedle M. Almost sure asymptotic stability of stochastic Volterra integro-differential equations with fading perturbations. Stoch. Anal. Appl.,2006,24(4):813-826.
[143]Luo J. Comparison principle and stability of Ito stochastic differential delay equations with Pois-son jump and Markovian switching. Nonlinear Anal.,2006,64(2):253-262.
[144]Luo J. A note on exponential stability in pth mean of solutions of stochastic delay differential equations. J. Comput. Appl. Math.,2007,198(1):143-148.
[145]Mao X. Attraction, stability and boundedness for stochastic differential delay equations. Nonlinear Anal.,2001,47(7):4795-4806.
[146]Mao X, Riedle M. Mean square stability of stochastic Volterra integro-differential equations. Systems Control Lett.,2006,55(6):459-465.
[147]Rodkina A, Basin M. On delay-dependent stability for a class of nonlinear stochastic delay-differential equations. Math Control Signals Syst.,2006,18(2):187-197.
[148]Mao X, Shaikhet L. Delay-dependent stability criteria for stochastic differential delay equations with markovian switching. Stab. Control Theory Appl.,2000,3(2):88-102.
[149]Arkin A, Ross J, McAdams H H. Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells. Genetics,1998,149(4):1633-1648.
[150]Gonze D, Halloy J, Goldbeter A. Robustness of circadian rhythms with respect to molecular noise. Proc. Natl. Acad. Sci.,2002,99(2):673-678.
[2]Parsons D J, Benjamin L R, Clarke J, et al. Weed Manager—A model-based decision support system for weed management in arable crops. Comput. Electron. Agric.,2009,65(2):155-167.
[3]Mortimer D, Dayan P, Burrage K, et al. Optimizing chemotaxis by measuring unbound-bound transitions. Physica D:Nonlinear Phenom.,2009,239(9):477-484.
[4]Allen L J S, Flores D A, Ratnayake R K, et al. Discrete-time deterministic and stochastic models for the spread of rabies. Appl. Math. Comput.,2002,132(2-3):271-292.
[5]Golightly A, Wilkinson D J. Bayesian inference for nonlinear multivariate diffusion models ob-served with error. Comput. Statistics Data Anal.,2008,52:1674-1693.
[6]Martin C F, Allen L J S. Urn Model Simulations of a Sexually Transmitted Disease Epidemic. Appl. Math. Comput.,1995,71(2-3):179-199.
[7]Hofmann N, Platen E, Schweizer M. Option pricing under incompleteness and stochastic volatility. Math. Finance,1992,2(3):153-187.
[8]Harris C J. Poisson Runge-Kutta methods for chemical reaction systems. in:Brebbi C A, editor, Proceedings of Mathematical Models and Environmental Problems, London:Pentech Press,1976, 269-282.
[9]Oksendal B. Stochastic differential equations:An introduction with applications. Berlin:Springer-Verlag,2003.
[10]Carletti M. Stochastic modelling of biological processes:[PhD Dissertation]. Brisbane, Australia: The University of Queensland,2008.
[11]Hasty J, Isaacs F, Dolnik M, et al. Designer gene networks:Towards fundamental cellular control. Chaos,2001,11(1):207-220.
[12]Burrage K, Tian T. Poisson Runge-Kutta methods for chemical reaction systems. in:Lu Y, Sun W, Tang T, editors, Proceedings of Advances in Scientific Computing and Applications, Beijing/New York:Science Press,2004,82-96.
[13]Tomioka R, Kimura H, Kobayashi T J, et al. Multivariate analysis of noise in genetic regulatory networks. J. Theor. Biol.,2004,229(4):501-521.
[14]Gillespie D T. The chemical Langevin equation. J. Chem. Phys.,2000,113(1):297-306.
[15]Gillespie D T. A rigorous derivation of the chemical master equation. Physica A,1992,188:404-425.
[16]Melykuti B, Burrage K, Zygalakis K C. Fast stochastic simulation of biochemical reaction sys-tems by alternative formulations of the chemical Langevin equation. J. Chem. Phys.,2010, 132(16):164109.
[17]Tian T, Burrage K. Bistability and switching in the lysis/lysogeny genetic regulatory network of bacteriophage A. J. Theor. Biol.,2004,227(2):229-237.
[18]Turner T E, Schnell S, Burrage K. Stochastic approaches for modelling in vivo reactions. Comput. Biol. Chem.,2004,28(3):165-178.
[19]Golightly A, Wilkinson D J. Bayesian inference for nonlinear multivariate diffusion models ob-served with error. Comput. Statistics Data Anal.,2008,52(3):1674-1693.
[20]Kesinger J C, Allen L J. Genetic models for plant pathosystems. Math. Biosci.,2002,177-178:247-269.
[21]Marquez-Lago T T, Leier A, Burrage K. Probability distributed time delays:integrating spatial effects into temporal models. BMC Syst. Biol.,2010,4(19):1-16.
[22]Bernard S, Cajavec B, Pujo-Menjouet L, et al. Modeling transcriptional feedback loops:The role of Gro/TLE1 in Hesl oscillations. Phil Transact A Math Phys. Eng. Sci.,2006,364(1842):1155-1170.
[23]Lewis J. Autoinhibition with transcriptional delay:a simple mechanism for the zebrafish somito-genesis oscillator. Curr. Biol.,2003,13(16):1398-1408.
[24]Hirata H, Yoshiura S, Ohtsuka T, et al. Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop. Science,2002,298(5594):840-843.
[25]Carletti M. Numerical solution of stochastic differential problems in the biosciences. J. Comput. Appl. Math.,2006,185(2):422-440.
[26]Carletti M. Mean-square stability of a stochastic model for bacteriophage infection with time delays. Math. Biosci.,2007,210(2):395-414.
[27]Carletti M. Numerical simulation of a Campbell-like stochastic delay model for bacteriophage infection. Math. Med. Biol.,2006,23(4):297-310.
[28]Tian T, Burrage K, Burrage P M, et al. Stochastic delay differential equations for genetic regulatory networks. J. Comput. Appl. Math.,2007,205(2):696-707.
[29]Mao X. Stochastic differential equations and their applications. Chichester:Horwood,1997.
[30]胡适耕,黄乘明,吴付科.随机微分方程.北京：科学出版社,2008.
[31]Ito K. Introduction to Probability Theory. Cambridge:Cambridge University Press,1984.
[32]Kunita H. Stochastic flows and stochastic differential equations. Cambridge:Cambridge Univer-sity Press,1990.
[33]Malliavin P. Stochastic analysis. Berlin:Springer,1997.
[34]Arnold L. Stochastic differential equations:theory and applications. New York:Wiley,1974.
[35]Sobcayk K. Stochastic differential equations, with applications to Physics and Engineering. Dor-drecht:Kluwer Academic Publishers,1991.
[36]Skorokhod A V. Studies in the theory of random processes. Reading, Mass:Addison-Wesley, 1965.
[37]Kloeden P E, Platen E. Numerical solution of stochastic differential equations. Berlin:Springer-Verlag,1992.
[38]Gard T C. Introduction to stochastic differential equations. New York:Marcel Dekker,1988.
[39]Platen E. An introduction to numerical methods for stochastic differential equations. Acta Numer., 1999,8:197-246.
[40]Milstein G N. Approximate integration of stochastic differential equations. Theory Prob. Appl., 1974,19(3):557-562.
[41]Platen E. Zur zeitdiskreten approximation von itoprozessen. Diss. B., Imath, Akad, der DDR, Berlin,1984.
[42]W.Rumelin. Numerical treatment of stochastic differential equations. SIAM J. Numer. Anal., 1982,19(3):604.
[43]Burrage K, Burrage P M. Order conditions of stochastic Runge-Kutta methods by B-series. SIAM J. Numer. Anal.,2000,38(5):1626-1646.
[44]王志勇.随机泛函微分方程的稳态数值解研究：[博士学位论文].武汉：华中科技大学,2008.
[45]Tian T, Burrage K. Implicit Taylor methods for stiff stochastic differential equations. Appl. Numer. Math.,2001,38(1-2):167-185.
[46]Burrage K, Tian T. Stiffly accurate Runge-Kutta methods for stiff stochastic differential equa-tions. Comput. Phys. Commun.,2001,142:186-190.
[47]Burrage K, Tian T. Implicit stochastic Runge-Kutta methods for stochastic differential equations. BIT Numer. Math.,2004,44(1):21-39.
[48]Debrabant K, Roβer A. Continuous weak approximation for stochastic differential equations. J. Comput. Appl. Math.,2008,214(1):259-273.
[49]Talay D. Second-order discretization schemes of stochastic differential systems for the computa-tion of the invariant law. Stochastics Stochastic Rep.,1990,29(1):13-36.
[50]Roβer A. Rooted tree analysis for order conditions of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations. Stochastic Anal. Appl.,2006,24(1):97-134.
[51]Komori Y. Weak first-or second-order implicit Runge-Kutta methods for stochastic differential equations with a scalar Wiener process. J. Comput. Appl. Math.,2008,217(1):166-179.
[52]Saito Y, Mitsui T. Stability analysis of numerical schemes for stochastic differential equations. SLAM J. Numer. Anal.,1996,33(6):2254-2267.
[53]Saito S, Mitsui T. T-stability of numerical scheme for stochastic differential equations. World Sci. Ser. Anal.,1993,2:333-344.
[54]Burrage K, Tian T. A note on the stability properties of the Euler methods for solving stochastic differential equations. New Zealand J. of Maths,2000,29:115-127.
[55]D.J.Higham, Mao X, Yuan C. Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. SAIM J. Numer. Anal.,2007,45(2):592-609.
[56]Pang S, Deng F, Mao X. Almost sure and moment exponential stability of Euler-Maruyama discretizations for hybrid stochastic differential equations. J. Comput. Appl. Math.,2008, 213(1):127-141.
[57]Hofmann N. Stability of weak numerical shemes for stochastic differential equations. Math. Comput. Simul.,1995,38(1-2):63-68.
[58]Hofmann N, Platen E. Stability of weak numerical shemes for stochastic differential equations. Comput. Math. Appl.,1994,28:45-47.
[59]Mohammed S. Stochastic functional differential equations. New York:Longman Scientific and Technical,1986.
[60]Ivanov A, Kazmerchuk Y, Swishchuk A. Theory, stochastic stability and applications of stochastic differential equations:a survey of recent results. Diff. Equa. Dyn. Syst.,2003,11(1-2):55-115.
[61]Longtin A. Stochastic delay-differential equations. in:Proceedings of Complex Time-delay Sys-tems. Springer Berlin,2009,177-195.
[62]Buckwar E. Introduction to the numerical analysis of stochastic delay differential equations. J. Comput. Appl. Math.,2000,125(1-2):297-307.
[63]Baker C T H, Buckwar E. Numerical analysis of explicit one-step methods for stochastic delay differential equations. LMS J. Comput. Math.,2000,3:315-335.
[64]Kuchler U, Platen E. Stong discrete time approximation of stochastic differential equations with time delay. Math. Comput. Simulat.,2000,54(1):189-205.
[65]Liu M, Cao W, Fan W. Convergence and stability of the semi-impplicit Euler method for a linear stochastic differential delay equation. J. Comput. Appl. Math.,2004,170(2):255-268.
[66]Buckwar E. One-step approximations for stochastic functional differential equations. Appl. Nu-mer. Math.,2006,56(5):667-681.
[67]覃婷婷.几类随机与延迟动力学系统的单步离散方法：[博士学位论文].武汉：华中科技大学,2010.
[68]Hu L, Gan S. Convergence and stability of the balanced methods for stochastic differential equa-tions with jumps. Int. J. Comput. Math.,2011,99999(1):1-20.
[69]Cao W, Liu M, Fan Z. MS-stability of the Euler-Maruyama method for stochastic differential delay equations. Appl. Math. Comput.,2004,159(1):127-135.
[70]Baker C T H, Buckwar E. Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations. J. Comput. Appl. Math.,2005, 184(2):404-427.
[71]曹婉容,刘明珠.随机延迟微分方程Euler-Maruyama数值方法的T-稳定性.哈尔滨工业大学学报,2005,37(3)：303-305.
[72]Wang Z, Zhang C. An analysis of stability of Milstein method for stochastic differential equations with delay. Comput. Math. Appl.,2006,51(9-10):1445-1452.
[73]王文强,黄山,李寿佛.非线性随机延迟微分方程Euler-Maruyama方法的均方稳定性.计算数学,2007,29(2)：217-224.
[74]王文强.几类非线性随机延迟微分方程数值方法的收敛性与稳定性：[博士学位论文].湘潭：湘潭大学,2007.
[75]王文强,黄山,李寿佛.非线性随机延迟微分方程半隐式Euler方法的收敛性.数值计算与计算机应用,2008,29(1)：73-80.
[76]王志勇,张诚坚.随机延迟微分方程的Milstein方法的非线性均方稳定性.应用数学,2008,21(1)：201-206.
[77]范振成.随机延迟微分方程Mil stein方法的均方稳定性.应用数学,2008,21(4)：743-747.
[78]Zhang H, Gan S, Hu L. The split-step backward Euler method for linear stochastic differential equations. J. Comput. Appl. Math.,2009,225(2):558-568.
[79]Wang X, Gan S. The improved split-step backward Euler method for stochastic differential delay equations. to be in Int. J. of Comput. Math.
[80]范振成.随机延迟微分方程的全隐式Euler方法.计算数学,2009,31(3)：287-298.
[81]王文强.非线性随机延迟微分方程Milstein方法的均方稳定性.系统仿真学报,2009,21(18)：5656-5658.
[82]Baker C, Buckwar E. Continuous θ-methods for the pantograph equation. Electron. Trans. Numer. Anal.,2000,11:131-151.
[83]Fan Z, Liu M, Cao W. Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations. J. Math. Anal. Appl.,2007,325(2):1142-1159.
[84]范振成,刘明珠.线性随机比例方程的渐近均方稳定性.应用数学,2007,20(3)：519-523.
[85]肖飞雁.几类随机延迟微分代数系统的数值分析：[博士学位论文].武汉：华中科技大学,2008.
[86]Golec J, Sathananthan S. Sample path approximation for stochastic integro-differential equations. Stoch. Anal. Appl.,1999,17(4):579-588.
[87]Golec J, Sathananthan S. Strong approximations of stochastic integro-differential equations. Dyn. Contin. Discrete Implus. Syst. Ser. B Appl. Algorithms,2001,8:139-151.
[88]Buckwar E. The theta-Maruyama scheme for stochastic functional differential equations with distributed memory term. Monte Carlo Methods Appl.,2004,10:235-244.
[89]Shaikhet L, Roberts J. Reliability of difference analogues to preserve stability properties of stochastic Volterra integro-differential equations. Adv. Difference Equ.,2006,2006(1):1-22.
[90]Ding X, Wu K, Liu M. Convergence and stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations. Int. J. Comput. Math.,2006,83(10):753-763.
[91]Rathinasamy A, Balachandran K. Mean-square stability of Milstein method for linear hybrid stochastic delay integro-differential equations. Nonlinear Anal. Hybrid Syst.,2008,2(4):1256-1263.
[92]Hairer E, Wanner G. Solving ordinary differential equations II:Stiff and differential-algebraic Problems. Berlin:Springer-Verlag,1996.
[93]Kloeden P E, Platen E. Higher-order implicit strong numerical scheme for stochastic differential equations. J. Stat. Phys.,2001,66(1):283-314.
[94]Milstein G N, Platen E, Schurz H. Balanced implicit methods for stiff stochastic systems. SIAM J. Numer. Anal.,1998,35(3):1010-1019.
[95]Alcock J, Burrage K. A note on the balanced method. BIT,2006,46(4):689-710.
[96]Brugnano L, K.Burrage, M.Burrage P. Adams-Type methods for the numerical solution of stochas-tic ordinary differential equations. BIT,2000,40(3):451-470.
[97]Burrage K, Tian T. Predictor-corrector methods of Runge-Kutta type for stochastic differential equations. SIAM J. Numer. Anal.,2002,40(4):1516-1537.
[98]Bruti-Liberati N, Platen E. Strong predictor-corrector Euler methods for stochastic differential equations. Stoch. Dynam.,2008,8(3):561-581.
[99]Elowitz M B, Leibler S. A synthetic oscillatory network of transcriptional regulators. Nature, 2000,403(6767):335-338.
[100]Sano J. Random monoallelic expression of three genes clustered within 60 kb of mouse t complex genomic DNA. Genome Res.,2001,11(11):1833-1841.
[101]Federoff N, Fontana W. Small numbers of big molecules. Science,2002,297(5584):1129-1131.
[102]Hume D A. Probability in transcriptional regulation and its implications for leukocyte differentia-tion and inducible gene expression. Blood,2000,96(7):2323-2328.
[103]Kepler T, Elston T. Stochasticity in transcriptional regulation:origins, consequences, and mathe-matical representations. Biophys. J.,2001,81 (6):3116-3136.
[104]McAdams H H, Arkin A. It's a noisy business! Genetic regulation at the nanomolar scale. Trends Genet.,1999,15(2):65-69.
[105]Anderson D F. A modified next reaction method for simulating chemical systems with time de-pendent propensities and delays. J. Chem. Phys.,2007,127(21):4107.
[106]Gibson M A, Bruck J. Efficient exact stochastic simulation of chemical systems with many species and many channels. J. Phys. Chem.,2000,104(9):1876-1889.
[107]Gillespie D T. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem.,1977, 81(25):2340-2361.
[108]Gillespie D T. Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys.,2001,115(4):1716-1733.
[109]Tian T, Burrage K. Binomial leap methods for simulating stochastic chemical kinetics. J. Chem. Phys.,2004,121 (21):10356-10364.
[110]Jensen M H, Sneppen K, Tiana G. Sustained oscillations and time delays in gene expression of protein Hesl. FEBS Lett.,2003,541 (1):176-177.
[111]Monk N A M. Oscillatory expression of Hesl, p53, and NF-jB driven by transcriptional time delays. Curr. Biol.,2003,13(16):1409-1413.
[112]Beretta E, Carletti M, Solimano F. On the effects of environmental fluctuations in a simple model of bacteria-bacteriophage interaction. Canad. Appl. Math. Quart.,2000,8(4):321-366.
[113]Cooke K, Driessche P, Zou X. Interaction of maturation delay and nonlinear birth in population and epidemic model. J. Math. Biol.,1999,39(4):332-352.
[114]Liu S, Beretta E. A stage-structured predator-prey model of Beddington-Deangelis type. SIAM J. Appl. Math.,2006,66(4):1101-1129.
[115]Chen L, Wang R, Zhou T. Noise-induced cooperative behavior in a multi-cell system. Bioinfor-matics,2005,21(11):2722-2729.
[116]Barrio M, K B, Leier A, et al. Oscillatory regulation of Hesl:discrete stochastic delay modelling and simulation. PLoS Comp. Biol.,2006,2(9):1017-1030.
[117]Bratsun D, Volfson D, Tsimring L S, et al. Delay-induced stochastic oscillations in gene regulation. in:Proceedings of Proc. Natl. Acad. Sci. U.S.A. National Academy of Sciences,2005,14593-14598.
[118]Cai X. Exact stochastic simulation of coupled chemical reactions with delays. J. Chem. Phys., 2007,126(12):4108.
[119]Haseltine E L, Rawlings J B. Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys.,2002,117(15):6959-6969.
[120]Rao C, Arkin A. Stochastic chemical kinetics and the quasi-steady-state assumption:application to the Gillespie algorithm. J. Chem. Phys.,2003,118(11):4999-5010.
[121]Burrage P M. Vectorised simulations for stochastic differential equations. ANZIAM J.,2004, 45(E):C350-C363.
[122]Kloeden P E, Platen E. Numerical solution of stochastic differential equations. Berlin:Springer, 1995.
[123]Burrage K, Burrage P M. High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations. Appl. Numer. Math.,1996,22(1-3):81-101.
[124]Burrage P M. Runge-Kutta methods for stochastic differential equations:[PhD Dissertation]. Queensland, Australia:The University of Queensland,1999.
[125]J.Higham D. An algorithmic introduction to numerical simulation of stochastic differential equa-tions. SIAM Review,2001,43(3):525-546.
[126]Milstein G N. Numerical integration of stochastic differential equations. Dordrecht:Kluwer Academic Publisher,1995.
[127]J.Higham D. Mean-square and asymptotic stability of numerical methods for stochastic ordinary differential equations. SIAM J. Numer. Anal.,2000,38(3):753-769.
[128]Buckwar E, Winkler R. Multistep methods for SDEs and their application to problems with small noise. SAIM J. Numer. Anal.,2006,44(2):779-803.
[129]Carletti M, Burrage K, Burrage P M. Numerical simulation of stochastic ordinary differential equations in biomathematical modelling. Math. Comput. Simul.,2004,64(2):271-277.
[130]Xu Y, Allena L J S, Perelson A S. Stochastic model of an influenza epidemic with drug resistance. J. Theor. Biol.,2007,248(1):179-193.
[131]Cao Y, Samuels D. Discrete stochastic simulation methods for chemically reacting systems. Meth-ods Enzymol.,2009,454:115-140.
[132]Higham D J. Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal.,2000,38(3):753-769.
[133]Bryden A, Higham D J. On the boundedness of asymptotic stability regions for the stochastic theta method. BIT Numer. Math.,2003,43(1):1-6.
[134]Higham D J, Mao X, Stuart A M. Exponential mean-square stability of numerical solutions to stochastic differential equations. LMS J. Comput. Math.,2003,6:297-313.
[135]Niu Y, Zhang C, Burrage K. Strong predictor-corrector approximations for SDDEs of Ito-type. submitted.
[136]Mao X, Sabanis S. Numerical solutions of stochastic differential delay equations under local Lipschitz conditions. J. Comp. Appl. Math.,2003,151(1):215-227.
[137]Hu Y, Mohammed S E A, Yan F. Discrete-time approximations of stochastic delay equations:the Milstein scheme. Ann. Probab.,2004,32(1):265-314.
[138]Lei J, Mackey M. Stochastic differential delay equation, moment stability, and application to hematopoietic stem cell regulation system. SIAM J. Appl. Math.,2007,67(2):387-407.
[139]Ortalo-Magne F, Rady S. Heterogeneity within communities:A stochastic model with tenure choice. J. Urban Economics,2008,64(1):1-17.
[140]Tass P. Phase resetting in medicine and biology:Stochastic modelling and data analysis. Berlin: Springer Verlag,1999.
[141]Thomson D J. A stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence, and its application to the problem of concentration variance. J. Fluid Mech., 1990,210:113-153.
[142]Appleby J A D, Riedle M. Almost sure asymptotic stability of stochastic Volterra integro-differential equations with fading perturbations. Stoch. Anal. Appl.,2006,24(4):813-826.
[143]Luo J. Comparison principle and stability of Ito stochastic differential delay equations with Pois-son jump and Markovian switching. Nonlinear Anal.,2006,64(2):253-262.
[144]Luo J. A note on exponential stability in pth mean of solutions of stochastic delay differential equations. J. Comput. Appl. Math.,2007,198(1):143-148.
[145]Mao X. Attraction, stability and boundedness for stochastic differential delay equations. Nonlinear Anal.,2001,47(7):4795-4806.
[146]Mao X, Riedle M. Mean square stability of stochastic Volterra integro-differential equations. Systems Control Lett.,2006,55(6):459-465.
[147]Rodkina A, Basin M. On delay-dependent stability for a class of nonlinear stochastic delay-differential equations. Math Control Signals Syst.,2006,18(2):187-197.
[148]Mao X, Shaikhet L. Delay-dependent stability criteria for stochastic differential delay equations with markovian switching. Stab. Control Theory Appl.,2000,3(2):88-102.
[149]Arkin A, Ross J, McAdams H H. Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells. Genetics,1998,149(4):1633-1648.
[150]Gonze D, Halloy J, Goldbeter A. Robustness of circadian rhythms with respect to molecular noise. Proc. Natl. Acad. Sci.,2002,99(2):673-678.