Convergence and stability of split-step θ methods for stochastic variable delay differential equations

References

• A. Abdulle and S. Cirilli, S-ROCK: Chebyshev methods for stiff SDEs, SIAM J. Sci. Comput. 30 (2008), pp. 997–1014.
   Web of Science ®Google Scholar
• A. Abdulle and T.J. Li, S-ROCK methods for stiff it o^ SDEs, Commun. Math. Sci. 6 (2008), pp. 845–868.
   Web of Science ®Google Scholar
• W.R. Cao, P. Hao, and Z.Q. Zhang, Split-step θ method for stochastic delay differential equations, Appl. Numer. Math. 76 (2014), pp. 19–33.
   Web of Science ®Google Scholar
• X.H. Ding, Q. Ma, and L. Zhang, Convergence and stability of the split-step θ-methods for stochastic differential equations, Comput. Math. Appl. 60 (2010), pp. 1310–1321.
   Web of Science ®Google Scholar
• M.A. Eissa, Y. Xiao, and B.P. Tian, Convergence and stability of two classes of theta methods with variable step size for a stochastic pantograph equations, J. Adv. Math. Appl. 5 (2016), pp. 95–106.
   Google Scholar
• Z.C. Fan, M.Z. Liu, and W.R. Cao, Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations, J. Math. Anal. Appl. 325 (2007), pp. 1142–1159.
   Web of Science ®Google Scholar
• Z.C. Fan and Y. Xiao, Stability of stochastic differential equation with unbounded delay, Ann. Differ. Equ. 3 (2009), pp. 309–315.
   Google Scholar
• D.J. Higham, X.R. Mao, and A.M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40 (2002), pp. 1041–1063.
   Web of Science ®Google Scholar
• C.M. Huang, Mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations, J. Comput. Appl. Math. 259 (2014), pp. 77–86.
   Web of Science ®Google Scholar
• F. Jiang, Y. Shen, and X.X. Liao, A note stability of the split-step backward Euler method for linear stochastic delay integro-differential equations, J. Sys. Sci. Complex. 25 (2012), pp. 873–879.
   Web of Science ®Google Scholar
• Q.Y. Li, S.Q. Gan, and X.J. Wang, Compensated stochastic theta methods for stochastic differential delay equations with jumps, Int. J. Comput. Math. 90 (2013), pp. 1057–1071.
   Web of Science ®Google Scholar
• G.Q. Liu and L. Zhang, Convergence of numerical solutions for semi-linear stochastic variable delay differential equations, J. Jilin Univ.(Sci. Ed.) 52 (2014), pp. 451–459. (in Chinese).
   Google Scholar
• M.Z. Liu, W.R Cao, and Z.C. Fan, Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equations, J. Comput. Appl. Math. 170 (2004), pp. 255–268.
   Web of Science ®Google Scholar
• Y.L. Lu, M.H. Song, and M.Z. Liu, Convergence and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments, J. Comput. Appl. Math. 317 (2017), pp. 55–71.
   Web of Science ®Google Scholar
• J. Tan and H. Wang, Convergence and stability of the split-step backward Euler method for linear stochastic delay integro-differential equations, Math. Comput. Model. 51 (2010), pp. 504–515.
   Google Scholar
• P. Wang and Y. Li, Split-step forward methods for stochastic differential equations, J. Comput. Appl. Math. 233 (2010), pp. 2641–2651.
   Web of Science ®Google Scholar
• X.J. Wang, S.Q. Gan, and D.S. Wang, θ-Maruyama methods for nonlinear stochastic differential delay equations, Appl. Numer. Math. 98 (2015), pp. 38–58.
   Web of Science ®Google Scholar
• C.J. Zhang and Y. Xie, Backward Euler-Maruyama method applied to nonlinear hybrid stochastic differential equations with time-variable delay, Sci. China Math. 62 (2019), pp. 597–616.
   Web of Science ®Google Scholar
• H.M. Zhang, S.Q. Gan, and L. Hu, The split-step backward Euler method for linear stochastic delay differential equations, J. Comput. Appl. Math. 225 (2009), pp. 558–568.
   Web of Science ®Google Scholar
• H.Y. Zhang, Y. Xiao, and F.Y. Guo, Convergence and stability of a numerical method for nonlinear stochastic pantograph equations, J. Franklin Inst. 351 (2014), pp. 3089–3103.
   Web of Science ®Google Scholar
• G.H. Zhao and M.Z. Liu, Numerical methods for nonlinear stochastic delay differential equations with jumps, Appl. Math. Comput. 233 (2014), pp. 222–231.
   Web of Science ®Google Scholar