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A note on a sifting-type lemma
Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations
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 |
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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