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Zhao, Xueyan; Deng, Feiqi; Kuang, Shifang

Numerical schemes for stochastic differential equations with variable and distributed delays: the interpolation approach. (English) Zbl 1474.65020

Abstr. Appl. Anal. 2014, Article ID 565812, 11 p. (2014).
Summary: A kind of the Euler-Maruyama schemes in discrete forms for stochastic differential equations with variable and distributed delays is proposed. The linear interpolation method is applied to deal with the values of the solutions at the delayed instants. The assumptions of this paper on the coefficients and related parameters are somehow weaker than those imposed by the related past literature. The error estimations for the Euler-Maruyama schemes are given, which are proved to be the same as those for the fundamental Euler-Maruyama schemes.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
34K50 Stochastic functional-differential equations
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[1] Kloeden, P. E.; Platen, E., Numerical Solution of Stochastic Differential Equations. Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), 23 (1992), Berlin, Germany: Springer, Berlin, Germany · Zbl 0925.65261
[2] Komori, Y.; Saito, Y.; Mitsui, T., Some issues in discrete approximate solution for stochastic differential equations, Computers & Mathematics with Applications, 28, 10 12, 269-278 (1994) · Zbl 0810.65145 · doi:10.1016/0898-1221(94)00197-9
[3] Platen, E., An introduction to numerical methods for stochastic differential equations, Acta Numerica, 8, 195-244 (1999)
[4] Milstein, G. N., Numerical Integration of Stochastic Differential Equations. Numerical Integration of Stochastic Differential Equations, Mathematics and Its Applications, 313 (1995), Dordrecht, The Netherlands: Kluwer Academic, Dordrecht, The Netherlands · Zbl 0810.65144
[5] Saito, Y.; Mitsui, T., Stability analysis of numerical schemes for stochastic differential equations, SIAM Journal on Numerical Analysis, 33, 6, 2254-2267 (1996) · Zbl 0869.60052 · doi:10.1137/S0036142992228409
[6] Mao, X., Stochastic Differential Equations and Their Applications. Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics & Applications (1997), Chichester, UK: Horwood Publishing, Chichester, UK · Zbl 0892.60057
[7] Baker, C. T. H.; Buckwar, E., Numerical analysis of explicit one-step methods for stochastic delay differential equations, LMS Journal of Computation and Mathematics, 3, 315-335 (2000) · Zbl 0974.65008 · doi:10.1112/S1461157000000322
[8] Buckwar, E., Introduction to the numerical analysis of stochastic delay differential equations, Journal of Computational and Applied Mathematics, 125, 1-2, 297-307 (2000) · Zbl 0971.65004 · doi:10.1016/S0377-0427(00)00475-1
[9] Küchler, U.; Platen, E., Strong discrete time approximation of stochastic differential equations with time delay, Mathematics and Computers in Simulation, 54, 1-3, 189-205 (2000) · doi:10.1016/S0378-4754(00)00224-X
[10] Hu, Y.; Mohammed, S. E. A.; Yan, F., Numerical Solution of Stochastic Differential Systems with Memory (2001), Southern Illinois University
[11] Mao, X.; Sabanis, S., Numerical solutions of stochastic differential delay equations under local Lipschitz condition, Journal of Computational and Applied Mathematics, 151, 1, 215-227 (2003) · Zbl 1015.65002 · doi:10.1016/S0377-0427(02)00750-1
[12] Mao, X., Exponential stability of equidistant Euler-Maruyama approximations of stochastic differential delay equations, Journal of Computational and Applied Mathematics, 200, 1, 297-316 (2007) · Zbl 1114.65005 · doi:10.1016/j.cam.2005.11.035
[13] Zhang, H.; Gan, S.; Hu, L., The split-step backward Euler method for linear stochastic delay differential equations, Journal of Computational and Applied Mathematics, 225, 2, 558-568 (2009) · Zbl 1183.65007 · doi:10.1016/j.cam.2008.08.032
[14] Mao, X., Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions, Applied Mathematics and Computation, 217, 12, 5512-5524 (2011) · Zbl 1215.65015 · doi:10.1016/j.amc.2010.12.023
[15] Jiang, F.; Shen, Y.; Wu, F., A note on order of convergence of numerical method for neutral stochastic functional differential equations, Communications in Nonlinear Science and Numerical Simulation, 17, 3, 1194-1200 (2012) · Zbl 1257.65005 · doi:10.1016/j.cnsns.2011.08.013
[16] Zhao, G.; Song, M. H.; Liu, M. Z., Exponential stability of Euler-Maruyama solutions for impulsive stochastic differential equations with delay, Applied Mathematics and Computation, 215, 9, 3425-3432 (2010) · Zbl 1229.65031 · doi:10.1016/j.amc.2009.10.037
[17] Zhou, S.; Wu, F., Convergence of numerical solutions to neutral stochastic delay differential equations with Markovian switching, Journal of Computational and Applied Mathematics, 229, 1, 85-96 (2009) · Zbl 1168.65007 · doi:10.1016/j.cam.2008.10.013
[18] Jiang, F.; Shen, Y.; Hu, J., Stability of the split-step backward Euler scheme for stochastic delay integro-differential equations with Markovian switching, Communications in Nonlinear Science and Numerical Simulation, 16, 2, 814-821 (2011) · Zbl 1221.65016 · doi:10.1016/j.cnsns.2010.04.034
[19] Wei, M., Convergence of numerical solutions for variable delay differential equations driven by Poisson random jump measure, Applied Mathematics and Computation, 212, 2, 409-417 (2009) · Zbl 1172.65008 · doi:10.1016/j.amc.2009.02.041
[20] Wang, L.-s.; Xue, H., Convergence of numerical solutions to stochastic differential delay equations with Poisson jump and Markovian switching, Applied Mathematics and Computation, 188, 2, 1161-1172 (2007) · Zbl 1128.65010 · doi:10.1016/j.amc.2006.10.058
[21] Wang, L.-s.; Mei, C.; Xue, H., The semi-implicit Euler method for stochastic differential delay equations with jumps, Applied Mathematics and Computation, 192, 2, 567-578 (2007) · Zbl 1193.65008 · doi:10.1016/j.amc.2007.03.027
[22] Mao, X., Approximate solutions for a class of stochastic evolution equations with variable delays. II, Numerical Functional Analysis and Optimization, 15, 1-2, 65-76 (1994) · Zbl 0796.60068 · doi:10.1080/01630569408816550
[23] Milošević, M., Highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama method, Mathematical and Computer Modelling, 54, 9-10, 2235-2251 (2011) · Zbl 1235.65009 · doi:10.1016/j.mcm.2011.05.033
[24] Wei, M., Convergence analysis of semi-implicit Euler methods for solving stochastic equations with variable delays and random jump magnitudes, Journal of Computational and Applied Mathematics, 235, 8, 2569-2580 (2011) · Zbl 1221.65022 · doi:10.1016/j.cam.2010.11.009
[25] Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional-Differential Equations. Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99 (1993), New York, NY, USA: Springer, New York, NY, USA · Zbl 0787.34002
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