中立型随机比例微分方程的数值解的指数稳定性（英文）
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摘要
本文主要利用半鞅收敛定理,研究中立型随机比例微分方程的数值稳定性.该文建立了线性的和非线性的中立型随机比例微分方程新的细则,我们将证明,在线性增长条件下,欧拉方法可以保留中立型随机比例微分方程的几乎处处指数稳定性,并且反向的欧拉方法能保留非线性的中立型随机比例微分方程的几乎处处指数稳定性.
This paper mainly studies the numerical stability of neutral stochastic pantograph differential equations(NSPDEs), with semimartingale convergence theorem.This paper establishes the new rules of linear and nonlinear NSPDEs. We will prove that under linear growth conditions, the Euler-Maruyama(EM) method can retain almost surely exponential stability of the NSPDEs, and the backward EM method can retain almost surely exponential stability for the nonlinear NSPDEs.
This paper mainly studies the numerical stability of neutral stochastic pantograph differential equations(NSPDEs), with semimartingale convergence theorem.This paper establishes the new rules of linear and nonlinear NSPDEs. We will prove that under linear growth conditions, the Euler-Maruyama(EM) method can retain almost surely exponential stability of the NSPDEs, and the backward EM method can retain almost surely exponential stability for the nonlinear NSPDEs.
引文
[1]MILOSEVIC M.Highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama method[J].Mathematical and Computer Modelling,2011,54(9-10):2235-2251.
[2]WU F,MAO X.Numerical solutions of neutral stochastic functional differential equations[J].SIAMJournal on Numerical Analysis,2008,46(1):1821-1841.
[3]ZHOU S,FANG Z.Numerical approximation of nonlinear neutral stochastic functional differential equation[J].Journal of Applied Mathematics and Computing,2013,41(1-2):427-445.
[4]MAO X.Numerical solutions of stochatic differential delay equations under the generalized Khasminskii-typeconditions[J].Applied Mathematics and Computation,2011,217(12):5512-5524.
[5]ZHOU S,WU F.Convergence of numerical solutions to neutral stochastic delay differential equations with Markovian switching[J].Journal of Computational and Applied Mathematics,2009,229(1):85-96.
[6]LI R,LIU M,WANG P.Convergence of numerical solutions to stochastic pantograph equations with Markovian switching[J].Applied Mathematics and Computation,2009,215(1):414-422.
[7]FAN Z,LIU M,CAO W.Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations[J].Journal of Mathematical Analysis and Applications,2007,325(2):1142-1159.
[8]WU F,MAO X,SZPRUCH L.Almost surely exponential stability of numerical solutions for stochastic delay differential equations[J].Numerische Mathematik,2010,115(4):681-697.
[9]ZHOU S.Exponential stability of numerical solutions to neutal stochastic functional differential equation[J].Applied Mathematics and Computation,2015,266(5):441-461.
[10]ZHOU S.Almost surely exponential stability of numerical solutions to for stochastic pantograph equations[J].Abstract and Applied Analysis,2014,10(6):1-9.
[11]MAO X.Exponential Stability of Stochastic Differential Equation[M].New York:Marcel Dekker Inc,1994.
[12]ZHOU S,XIE S,FANG Z.Almost surely exponential stability of the backward Euler-Maruyama discretization for highly nonlinear stochastic functional differential equation[J].Applied Mathematics and Computation,2014,236(6):150-160.
[13]MAO X,SHEN Y,GRAY A.Almost surely exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equation[J].Journal of Computational and Applied Mathematic,2011,235(5):1213-1226.
[2]WU F,MAO X.Numerical solutions of neutral stochastic functional differential equations[J].SIAMJournal on Numerical Analysis,2008,46(1):1821-1841.
[3]ZHOU S,FANG Z.Numerical approximation of nonlinear neutral stochastic functional differential equation[J].Journal of Applied Mathematics and Computing,2013,41(1-2):427-445.
[4]MAO X.Numerical solutions of stochatic differential delay equations under the generalized Khasminskii-typeconditions[J].Applied Mathematics and Computation,2011,217(12):5512-5524.
[5]ZHOU S,WU F.Convergence of numerical solutions to neutral stochastic delay differential equations with Markovian switching[J].Journal of Computational and Applied Mathematics,2009,229(1):85-96.
[6]LI R,LIU M,WANG P.Convergence of numerical solutions to stochastic pantograph equations with Markovian switching[J].Applied Mathematics and Computation,2009,215(1):414-422.
[7]FAN Z,LIU M,CAO W.Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations[J].Journal of Mathematical Analysis and Applications,2007,325(2):1142-1159.
[8]WU F,MAO X,SZPRUCH L.Almost surely exponential stability of numerical solutions for stochastic delay differential equations[J].Numerische Mathematik,2010,115(4):681-697.
[9]ZHOU S.Exponential stability of numerical solutions to neutal stochastic functional differential equation[J].Applied Mathematics and Computation,2015,266(5):441-461.
[10]ZHOU S.Almost surely exponential stability of numerical solutions to for stochastic pantograph equations[J].Abstract and Applied Analysis,2014,10(6):1-9.
[11]MAO X.Exponential Stability of Stochastic Differential Equation[M].New York:Marcel Dekker Inc,1994.
[12]ZHOU S,XIE S,FANG Z.Almost surely exponential stability of the backward Euler-Maruyama discretization for highly nonlinear stochastic functional differential equation[J].Applied Mathematics and Computation,2014,236(6):150-160.
[13]MAO X,SHEN Y,GRAY A.Almost surely exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equation[J].Journal of Computational and Applied Mathematic,2011,235(5):1213-1226.