American Institute of Mathematical Sciences

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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 and 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

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