Strong convergence of the split-step theta method for stochastic delay differential equations with nonglobally Lipschitz continuous coefficients. (English) Zbl 1470.65011

Yue, Chao; Huang, Chengming

Abstr. Appl. Anal. 2014, Article ID 157498, 9 p. (2014).

Summary: This paper is concerned with the convergence analysis of numerical methods for stochastic delay differential equations. We consider the split-step theta method for nonlinear nonautonomous equations and prove the strong convergence of the numerical solution under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. In particular, these conditions admit that the diffusion coefficient is highly nonlinear. Furthermore, the obtained results are supported by numerical experiments.

MSC:

65C30	Numerical solutions to stochastic differential and integral equations
65L03	Numerical methods for functional-differential equations
34K50	Stochastic functional-differential equations
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35	Computational methods for stochastic equations (aspects of stochastic analysis)

References:

[1]	Lee, M.-K.; Kim, J.-H.; Kim, J., A delay financial model with stochastic volatility; Martingale method, Physica A: Statistical Mechanics and Its Applications, 390, 16, 2909-2919 (2011) · doi:10.1016/j.physa.2011.03.032
[2]	Mao, X., Stochastic Differential Equations and Applications (2008), Chichester, UK: Horwood Publishing Limited,, Chichester, UK · doi:10.1533/9780857099402
[3]	Tian, T.; Burrage, K.; Burrage, P. M. a., Stochastic delay differential equations for genetic regulatory networks, Journal of Computational and Applied Mathematics, 205, 2, 696-707 (2007) · Zbl 1112.92029 · doi:10.1016/j.cam.2006.02.063
[4]	Carletti, M., Numerical solution of stochastic differential problems in the biosciences, Journal of Computational and Applied Mathematics, 185, 2, 422-440 (2006) · Zbl 1077.65007 · doi:10.1016/j.cam.2005.03.020
[5]	Ivanov, A. F.; Swishchuk, A. V., Optimal control of stochastic differential delay equations with application in economics, International Journal of Qualitative Theory of Differential Equations and Applications, 2, 2, 201-213 (2008) · Zbl 1263.34116
[6]	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
[7]	Hu, Y.; Mohammed, S.-E. A.; Yan, F., Discrete-time approximations of stochastic delay equations: the milstein scheme, The Annals of Probability, 32, 1, 265-314 (2004) · Zbl 1062.60065 · doi:10.1214/aop/1078415836
[8]	Huang, C., Mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations, Journal of Computational and Applied Mathematics, 259, 77-86 (2014) · Zbl 1291.65020 · doi:10.1016/j.cam.2013.03.038
[9]	Fan, Z.; Liu, M.; Cao, W., Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations, Journal of Mathematical Analysis and Applications, 325, 2, 1142-1159 (2007) · Zbl 1107.60030 · doi:10.1016/j.jmaa.2006.02.063
[10]	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
[11]	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
[12]	Wang, X.; Gan, S., The improved split-step backward Euler method for stochastic differential delay equations, International Journal of Computer Mathematics, 88, 11, 2359-2378 (2011) · Zbl 1235.65010 · doi:10.1080/00207160.2010.538388
[13]	Liu, M.; Cao, W.; Fan, Z., Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation, Journal of Computational and Applied Mathematics, 170, 2, 255-268 (2004) · Zbl 1059.65006 · doi:10.1016/j.cam.2004.01.040
[14]	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
[15]	Bao, J.; Yuan, C., Convergence rate of EM scheme for SDDEs, Proceedings of the American Mathematical Society, 141, 9, 3231-3243 (2013) · Zbl 1277.65006 · doi:10.1090/S0002-9939-2013-11886-1
[16]	Wu, F.; Mao, X.; Chen, K., The Cox-Ingersoll-Ross model with delay and strong convergence of its Euler-Maruyama approximate solutions, Applied Numerical Mathematics, 59, 10, 2641-2658 (2009) · Zbl 1172.65009 · doi:10.1016/j.apnum.2009.03.004
[17]	Liu, J.; Zhou, J., Convergence rate of Euler-Maruyama scheme for stochastic pantograph differential equations, Communications in Nonlinear Science and Numerical Simulation, 19, 6, 1697-1705 (2014) · Zbl 1457.65006 · doi:10.1016/j.cnsns.2013.10.015
[18]	Mao, X.; Rassias, M. J., Khasminskii-type theorems for stochastic differential delay equations, Stochastic Analysis and Applications, 23, 5, 1045-1069 (2005) · Zbl 1082.60055 · doi:10.1080/07362990500118637
[19]	Higham, D. J.; Mao, X.; Stuart, A. M., Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal on Numerical Analysis, 40, 3, 1041-1063 (2002) · Zbl 1026.65003 · doi:10.1137/S0036142901389530
[20]	Higham, D. J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43, 3, 525-546 (2001) · Zbl 0979.65007 · doi:10.1137/S0036144500378302