American Institute of Mathematical Sciences

Advanced
February  2013, 33(2): 885-903. doi: 10.3934/dcds.2013.33.885

Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations

Fuke Wu 1, , Xuerong Mao 2, and  Peter E. Kloeden 3,

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074
2. 

Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, United Kingdom
3. 

Institut für Mathematik, Johann Wolfgang Goethe Universität, D-60054 Frankfurt am Main, Germany

Received  August 2011 Revised  December 2011 Published  September 2012

A discrete stochastic Razumikhin-type theorem is established to investigate whether the Euler--Maruyama (EM) scheme can reproduce the moment exponential stability of exact solutions of stochastic functional differential equations (SFDEs). In addition, the Chebyshev inequality and the Borel-Cantelli lemma are applied to show the almost sure stability of the EM approximate solutions of SFDEs. To show our idea clearly, these results are used to discuss stability of numerical solutions of two classes of special SFDEs, including stochastic delay differential equations (SDDEs) with variable delay and stochastically perturbed equations.
Keywords: stochastic functional differential equations, stochastically perturbed equations., Moment exponential stability, Razumikhin-type theorem, Euler--Maruyama method.
Mathematics Subject Classification: Primary: 65C20, 65C30; Secondary: 65Q0.
Citation: Fuke Wu, Xuerong Mao, Peter E. Kloeden. Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 885-903. doi: 10.3934/dcds.2013.33.885
References:
[1]

C. T. H. Baker and E. Buckwar, Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations, J. Comput. Appl. Math., 184 (2005), 404-427. doi: 10.1016/j.cam.2005.01.018.  Google Scholar

[2]

K. Burrage, P. Burrage and T. Mitsui, Numerical solutions of stochastic differential equations-implication and stability issues, J. Comput. Appl. Math., 125 (2000), 171-182. doi: 10.1016/S0377-0427(00)00467-2.  Google Scholar

[3]

K. Burrage and T. Tian, A note on the stability properties of the Euler methods for solving stochastic differential equations, New Zealand J. of Maths., 29 (2000), 115-127.  Google Scholar

[4]

T. Caraballo, P. E. Kloeden and J. Real, Discretization of asymptotically stable stationary solutions of delay differential equations with a random stationary delay, J. Dyn. & Diff. Eqns., 18 (2006), 863-880. doi: 10.1007/s10884-006-9022-5.  Google Scholar

[5]

J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations,'' Springer, 1993.  Google Scholar

[6]

D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal., 38 (2000), 753-769. doi: 10.1137/S003614299834736X.  Google Scholar

[7]

D. J. Higham, X. Mao and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal., 45 (2007), 592-609. doi: 10.1137/060658138.  Google Scholar

[8]

B. Liu and H. J. Marquez, Razumikhin-type stability theorems for discrete delay systems, Automatica, 43 (2007), 1219-1225. doi: 10.1016/j.automatica.2006.12.032.  Google Scholar

[9]

X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Sto. Proc. Their Appl., 65 (1996), 233-250. doi: 10.1016/S0304-4149(96)00109-3.  Google Scholar

[10]

X. Mao, "Stochastic Differential Equations and Their Applications,'' Horwood Publishing, Chichester, England, 1997.  Google Scholar

[11]

X. Mao, Numerical solutions of stochastic functional differential equations, LMS J. Comput. Math., 6 (2003), 141-161.  Google Scholar

[12]

S. Pang, F. Deng and X. Mao, Almost sure and moment exponential stability of Euler-Maruyama discretizations for hybrid stochastic differential equations, J. Comput. Appl. Math., 213 (2008), 127-141. doi: 10.1016/j.cam.2007.01.003.  Google Scholar

[13]

Y. Saito and T. Mitsui, T-stability of numerical scheme for stochastic differential equations, World Sci. Ser. Appl. Anal., 2 (1993), 333-344.  Google Scholar

[14]

Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267. doi: 10.1137/S0036142992228409.  Google Scholar

[15]

F. Wu and X. Mao, Numerical solutions of neutral stochastic functional differential equations, SIAM J. Numer. Anal., 46 (2008), 1821-1841. doi: 10.1137/070697021.  Google Scholar

[16]

F. Wu, X. Mao and L. Szpruch, Almost sure exponential stability of numerical solutions for stochastic delay differential equations, Numer. Math., 115 (2010), 681-697. doi: 10.1007/s00211-010-0294-7.  Google Scholar

[17]

F. Wu, X. Mao and P. E. Kloeden, Almost sure exponential stability of the Euler-Maruyama approximations for stochastic functional differential equations, Random Oper. Stoch. Equ., 19 (2011), 165-186. doi: 10.1515/ROSE.2011.010.  Google Scholar

[18]

C. Zhang and S. Vandewalle, Stability analysis of Volterra delay-integro-differential equations and their backward differentiation time discretization, J. comput. Appl. Math., 164-165 (2004), 797-814.  Google Scholar

[19]

S. Zhang and M. P. Chen, A new Razumikhin Theorem for delay difference equations, Comput. Math. Appl., 36 (1998), 405-412. doi: 10.1016/S0898-1221(98)80041-2.  Google Scholar

show all references

References:
[1]

C. T. H. Baker and E. Buckwar, Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations, J. Comput. Appl. Math., 184 (2005), 404-427. doi: 10.1016/j.cam.2005.01.018.  Google Scholar

[2]

K. Burrage, P. Burrage and T. Mitsui, Numerical solutions of stochastic differential equations-implication and stability issues, J. Comput. Appl. Math., 125 (2000), 171-182. doi: 10.1016/S0377-0427(00)00467-2.  Google Scholar

[3]

K. Burrage and T. Tian, A note on the stability properties of the Euler methods for solving stochastic differential equations, New Zealand J. of Maths., 29 (2000), 115-127.  Google Scholar

[4]

T. Caraballo, P. E. Kloeden and J. Real, Discretization of asymptotically stable stationary solutions of delay differential equations with a random stationary delay, J. Dyn. & Diff. Eqns., 18 (2006), 863-880. doi: 10.1007/s10884-006-9022-5.  Google Scholar

[5]

J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations,'' Springer, 1993.  Google Scholar

[6]

D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal., 38 (2000), 753-769. doi: 10.1137/S003614299834736X.  Google Scholar

[7]

D. J. Higham, X. Mao and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal., 45 (2007), 592-609. doi: 10.1137/060658138.  Google Scholar

[8]

B. Liu and H. J. Marquez, Razumikhin-type stability theorems for discrete delay systems, Automatica, 43 (2007), 1219-1225. doi: 10.1016/j.automatica.2006.12.032.  Google Scholar

[9]

X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Sto. Proc. Their Appl., 65 (1996), 233-250. doi: 10.1016/S0304-4149(96)00109-3.  Google Scholar

[10]

X. Mao, "Stochastic Differential Equations and Their Applications,'' Horwood Publishing, Chichester, England, 1997.  Google Scholar

[11]

X. Mao, Numerical solutions of stochastic functional differential equations, LMS J. Comput. Math., 6 (2003), 141-161.  Google Scholar

[12]

S. Pang, F. Deng and X. Mao, Almost sure and moment exponential stability of Euler-Maruyama discretizations for hybrid stochastic differential equations, J. Comput. Appl. Math., 213 (2008), 127-141. doi: 10.1016/j.cam.2007.01.003.  Google Scholar

[13]

Y. Saito and T. Mitsui, T-stability of numerical scheme for stochastic differential equations, World Sci. Ser. Appl. Anal., 2 (1993), 333-344.  Google Scholar

[14]

Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267. doi: 10.1137/S0036142992228409.  Google Scholar

[15]

F. Wu and X. Mao, Numerical solutions of neutral stochastic functional differential equations, SIAM J. Numer. Anal., 46 (2008), 1821-1841. doi: 10.1137/070697021.  Google Scholar

[16]

F. Wu, X. Mao and L. Szpruch, Almost sure exponential stability of numerical solutions for stochastic delay differential equations, Numer. Math., 115 (2010), 681-697. doi: 10.1007/s00211-010-0294-7.  Google Scholar

[17]

F. Wu, X. Mao and P. E. Kloeden, Almost sure exponential stability of the Euler-Maruyama approximations for stochastic functional differential equations, Random Oper. Stoch. Equ., 19 (2011), 165-186. doi: 10.1515/ROSE.2011.010.  Google Scholar

[18]

C. Zhang and S. Vandewalle, Stability analysis of Volterra delay-integro-differential equations and their backward differentiation time discretization, J. comput. Appl. Math., 164-165 (2004), 797-814.  Google Scholar

[19]

S. Zhang and M. P. Chen, A new Razumikhin Theorem for delay difference equations, Comput. Math. Appl., 36 (1998), 405-412. doi: 10.1016/S0898-1221(98)80041-2.  Google Scholar

[1]

Fuke Wu, George Yin, Le Yi Wang. Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching. Mathematical Control & Related Fields, 2015, 5 (3) : 697-719. doi: 10.3934/mcrf.2015.5.697
[2]

Weiyin Fei, Liangjian Hu, Xuerong Mao, Dengfeng Xia. Advances in the truncated Euler–Maruyama method for stochastic differential delay equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2081-2100. doi: 10.3934/cpaa.2020092
[3]

Wei Mao, Liangjian Hu, Xuerong Mao. Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 587-613. doi: 10.3934/dcdsb.2018198
[4]

Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang. pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 209-226. doi: 10.3934/dcdsb.2017011
[5]

Wei Mao, Liangjian Hu, Xuerong Mao. Razumikhin-type theorems on polynomial stability of hybrid stochastic systems with pantograph delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3217-3232. doi: 10.3934/dcdsb.2020059
[6]

Fuke Wu, Shigeng Hu. The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 1065-1094. doi: 10.3934/dcds.2012.32.1065
[7]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169
[8]

Marat Akhmet, Duygu Aruğaslan. Lyapunov-Razumikhin method for differential equations with piecewise constant argument. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 457-466. doi: 10.3934/dcds.2009.25.457
[9]

Minghui Song, Liangjian Hu, Xuerong Mao, Liguo Zhang. Khasminskii-type theorems for stochastic functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1697-1714. doi: 10.3934/dcdsb.2013.18.1697
[10]

Yufeng Shi, Qingfeng Zhu. A Kneser-type theorem for backward doubly stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1565-1579. doi: 10.3934/dcdsb.2010.14.1565
[11]

Leonid Shaikhet. Stability of equilibriums of stochastically perturbed delay differential neoclassical growth model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1565-1573. doi: 10.3934/dcdsb.2017075
[12]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[13]

Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2923-2938. doi: 10.3934/dcdsb.2017157
[14]

Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103
[15]

Zhen Wang, Xiong Li, Jinzhi Lei. Second moment boundedness of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2963-2991. doi: 10.3934/dcdsb.2014.19.2963
[16]

Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure & Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016
[17]

Ya Wang, Fuke Wu, Xuerong Mao, Enwen Zhu. Advances in the LaSalle-type theorems for stochastic functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 287-300. doi: 10.3934/dcdsb.2019182
[18]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687
[19]

Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157
[20]

Tomás Caraballo, Carlos Ogouyandjou, Fulbert Kuessi Allognissode, Mamadou Abdoul Diop. Existence and exponential stability for neutral stochastic integro–differential equations with impulses driven by a Rosenblatt process. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 507-528. doi: 10.3934/dcdsb.2019251

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (17)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences