%0 Journal Article %T Two-step Maruyama schemes for nonlinear stochastic differential delay equations %A Lei, Dongxia %A Zong, Xiaofeng %A Hu, Junhao %J Journal of Nonlinear Sciences and Applications %D 2017 %V 10 %N 10 %@ ISSN 2008-1901 %F Lei2017 %X This work concerns the two-step Maruyama schemes for nonlinear stochastic differential delay equations (SDDEs). We first examine the strong convergence rates of the split two-step Maruyama scheme and linear two-step Maruyama scheme (including Adams-Bashforth and Adams-Moulton schemes) for nonlinear SDDEs with highly nonlinear delay variables, then we investigate the exponential mean square stability and exponential decay rates of the two classes of two-step Maruyama schemes. These results are important for three reasons: first, the convergence rates are established under the non-global Lipschitz condition; second, these stability results show that these two-step Maruyama schemes can not only reproduce the exponential mean square stability, but also preserve the bound of Lyapunov exponent for sufficient small stepsize; third, they are also suitable for the corresponding two-step Maruyama methods of stochastic ordinary differential equations (SODEs). %9 journal article %R 10.22436/jnsa.010.10.11 %U http://dx.doi.org/10.22436/jnsa.010.10.11 %P 5245--5260 %0 Journal Article %T Testing continuous-time models of the spot interest rate %A Y. Ait-Sahalia %J Rev. Financial Stud. %D 1996 %V 9 %F Ait-Sahalia1996 %0 Journal Article %T Convergence rate of EM scheme for SDDEs %A J.-H. Bao %A C.-G. Yuan %J Proc. Amer. Math. Soc. %D 2013 %V 141 %F Bao2013 %0 Journal Article %T Stability of epidemic model with time delays influenced by stochastic perturbations %A E. Beretta %A V. Kolmanovskii %A L. Shaikhet %J Delay systems, Lille, (1996), Math. Comput. Simulation %D 1998 %V 45 %F Beretta1998 %0 Journal Article %T On the boundedness of asymptotic stability regions for the stochastic theta method %A A. Bryden %A D. J. Higham %J BIT %D 2003 %V 43 %F Bryden2003 %0 Journal Article %T Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations %A E. Buckwar %A R. Horvath-Bokor %A R. Winkler %J BIT %D 2006 %V 46 %F Buckwar2006 %0 Journal Article %T On two-step schemes for SDEs with small noise %A E. Buckwar %A R. Winkler %J PAMM %D 2004 %V 4 %F Buckwar2004 %0 Journal Article %T Multistep methods for SDEs and their application to problems with small noise %A E. Buckwar %A R. Winkler %J SIAM J. Numer. Anal. %D 2006 %V 44 %F Buckwar2006 %0 Journal Article %T Multi-step Maruyama methods for stochastic delay differential equations %A E. Buckwar %A R. Winkler %J Stoch. Anal. Appl. %D 2007 %V 25 %F Buckwar2007 %0 Journal Article %T On exponential mean-square stability of two-step Maruyama methods for stochastic delay differential equations %A W.-R. Cao %A Z.-Q. Zhang %J J. Comput. Appl. Math. %D 2013 %V 245 %F Cao2013 %0 Journal Article %T Noise-induced transitions in human postural sway %A C. W. Eurich %A J. G. Milton %J Phys. Rev. E %D 1996 %V 54 %F Eurich1996 %0 Journal Article %T Mean-square and asymptotic stability of the stochastic theta method %A D. J. Higham %J SIAM J. Numer. Anal. %D 2000 %V 38 %F Higham2000 %0 Journal Article %T Strong convergence of Euler-type methods for nonlinear stochastic differential equations %A D. J. Higham %A X.-R. Mao %A A. M. Stuart %J SIAM J. Numer. Anal. %D 2002 %V 40 %F Higham2002 %0 Journal Article %T Exponential mean-square stability of numerical solutions to stochastic differential equations %A D. J. Higham %A X.-R. Mao %A A. M. Stuart %J LMS J. Comput. Math. %D 2003 %V 6 %F Higham2003 %0 Journal Article %T Complete models with stochastic volatility %A D. G. Hobson %A L. C. G. Rogers %J Math. Finance %D 1998 %V 8 %F Hobson1998 %0 Journal Article %T Exponential mean square stability of numerical methods for systems of stochastic differential equations %A C.-M. Huang %J J. Comput. Appl. Math. %D 2012 %V 236 %F Huang2012 %0 Journal Article %T Mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations %A C.-M. Huang %J J. Comput. Appl. Math. %D 2014 %V 259 %F Huang2014 %0 Journal Article %T Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients %A M. Hutzenthaler %A A. Jentzen %J Mem. Amer. Math. Soc. %D 2015 %V 236 %F Hutzenthaler2015 %0 Book %T Numerical solution of stochastic differential equations %A P. E. Kloeden %A E. Platen %D 1992 %I Applications of Mathematics (New York), Springer-Verlag %C Berlin %F Kloeden1992 %0 Journal Article %T Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation %A M.-Z. Liu %A W.-R. Cao %A Z.-C. Fan %J J. Comput. Appl. Math. %D 2004 %V 170 %F Liu2004 %0 Journal Article %T Noise and critical behavior of the pupil light reflex at oscillation onset %A A. Longtin %A J. G. Milton %A J. E. Bos %A M. C. Mackey %J Phys. Rev. A %D 1990 %V 41 %F Longtin1990 %0 Book %T Stochastic differential equations and their applications %A X.-R. Mao %D 1997 %I Horwood Publishing Series in Mathematics & Applications, Horwood Publishing Limited %C Chichester %F Mao1997 %0 Book %T Numerical integration of stochastic differential equations %A G. N. Milstein %D 1995 %I Translated and revised from the 1988 Russian original, Mathematics and its Applications, Kluwer Academic Publishers Group %C Dordrecht %F Milstein1995 %0 Book %T Stochastic numerics for mathematical physics %A G. N. Milstein %A M. V. Tretyakov %D 2004 %I Scientific Computation, Springer- Verlag %C Berlin %F Milstein2004 %0 Journal Article %T Stability analysis of numerical schemes for stochastic differential equations %A Y. Saito %A T. Mitsui %J SIAM J. Numer. Anal. %D 1996 %V 33 %F Saito1996 %0 Journal Article %T Mean-square convergence of stochastic multi-step methods with variable step-size %A T. Sickenberger %J J. Comput. Appl. Math. %D 2008 %V 212 %F Sickenberger2008 %0 Journal Article %T Effects of environmental fluctuation on plankton allelopathy %A P. K. Tapaswi %A A. Mukhopadhyay %J J. Math. Biol. %D 1999 %V 39 %F Tapaswi1999 %0 Journal Article %T Asymptotic mean-square stability of two-step Maruyama schemes for stochastic differential equations %A A. Tocino %A M. J. Senosiain %J J. Comput. Appl. Math. %D 2014 %V 260 %F Tocino2014 %0 Journal Article %T The improved split-step backward Euler method for stochastic differential delay equations %A X.-J. Wang %A S.-Q. Gan %J Int. J. Comput. Math. %D 2011 %V 88 %F Wang2011 %0 Journal Article %T A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise %A X.-J. Wang %A S.-Q. Gan %A D.-S. Wang %J BIT %D 2012 %V 52 %F Wang2012 %0 Journal Article %T The Cox-Ingersoll-Ross model with delay and strong convergence of its Euler-Maruyama approximate solutions %A F.-K. Wu %A X.-R. Mao %A K. Chen %J Appl. Numer. Math. %D 2009 %V 59 %F Wu2009 %0 Journal Article %T Choice of \(\theta\) and mean-square exponential stability in the stochastic theta method of stochastic differential equations %A X.-F. Zong %A F.-K. Wu %J J. Comput. Appl. Math. %D 2014 %V 255 %F Zong2014 %0 Journal Article %T Convergence and stability of the semi-tamed Euler scheme for stochastic differential equations with non-Lipschitz continuous coefficients %A X.-F. Zong %A F.-K. Wu %A C.-M. Huang %J Appl. Math. Comput. %D 2014 %V 228 %F Zong2014 %0 Journal Article %T Preserving exponential mean square stability and decay rates in two classes of theta approximations of stochastic differential equations %A X.-F. Zong %A F.-K. Wu %A C.-M. Huang %J J. Difference Equ. Appl. %D 2014 %V 20 %F Zong2014 %0 Journal Article %T Theta-Euler schemes for SDEs with non-global Lipschitz continuous coeffcients %A X.-F. Zong %A F.-K. Wu %A C.-M. Huang %J %D Submitted %V %F ZongSubmitted