References
- Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation, report, 2005.J.A.D. Appleby and E. Buckwar
- Baker , C. T.H. and Buckwar , E. 2000 . Continuous θ-methods for the stochastidc pantograph equation . Electron. Trans. Numer. Anal. , 11 : 131 – 151 .
- Bellen , A. , Guglielmi , N. and Torelli , L. 1997 . Asymptotic stability properties of - methods for the pantograph equation . Appl. Numer. Math. , 24 : 279 – 293 .
- Brugnano , L. , Burrage , K. and Burrage , P. M. 2000 . Adams-type methods for the numerical solution of stochastic ordinary differential equations . 40 : 451 – 470 . BIT
- Buckwar , E. 2000 . Introduction to the numerical analysis of stochastic dilay differential equations . J. Comput. Appl. Math. , 125 : 297 – 307 .
- Fan , Z. C. , Liu , M. Z. and Cao , W. R. 2007 . Existence and uniqueness of the solutions and convergene of semi-implicit Euler methods for stochastic pantograph equations . J. Math. Anal. Appl. , 325 : 1142 – 1159 .
- Hale , J. K. 1993 . Introduction to Functional Differential Equations , New York : Springer .
- Higham , D. J. , Mao , X. R. and Stuart , A. M. 2002 . Strong convergence of Euler-type methods for nonlinear stochastic differential equations . SIAM J. Numer. Anal. , 40 : 1041 – 1063 .
- Hu , Y. and Mohammed , S. E.A. 2004 . Discrete-time approximation of stochastic delay equations: The Milstein scheme . Ann. Probab. , 32 : 265 – 314 .
- Iserles , A. 1993 . On the generalized pantograph functional–differential equation . European J. Appl. Math. , 4 : 1 – 38 .
- Iserles , A. 1994 . Numerical analysis of delay differential equations with variable dealys . Ann. Numer. Math. , 1 : 133 – 152 .
- Iserles , A. and Terjeki , J. 1995 . Stability and asymptotical stability of functional- defferential equations . J. London. Math. Soc. , 51 : 559 – 572 .
- Liu , Y. 1996 . On the θ-method for delay differential equations with infinite lag . J. Comput. Appl. Math. , 71 : 177 – 190 .
- Liu , Y. 1997 . Numerical investigation of the pantograph equation . Appl. Numer. Math. , 24 : 309 – 317 .
- Liu , M. Z. , Cao , W. R. and Fan , Z. C. 2004 . Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation . J. Comput. Appl. Math. , 170 : 255 – 268 .
- Liu , M. Z. , Yang , Z. W. and Hu , G. D. 2005 . Asymptotical stability of the numerical methods with the constant step size for the pantograph equation . 45 : 743 – 759 . BIT
- Liu , M. Z. , Fan , Z. C. and Song , M. H. 2009 . The αth moment stability for the stochastic pantograph equation . J. Comput. Appl. Math. , 233 : 109 – 120 .
- Mao , X. R. 1996 . Razumikhin-type theorems on exponential stability of stochastic functional differential equations . Stochast. Process. Appl. , 65 : 233 – 250 .
- Mohammed , S. E.A. 1984 . Stochastic Functional Differential Equations , London : Pitman . Research Notes in Mathematics
- Mao , X. R. 1997 . Stochastic Differential Equations and Applications , New York : Harwood .
- Wang , Z. Y. and Zhang , C. J. 2006 . An analysis of stability of Milstein method for SDEs with delay . Comput. Math. Appl. , 51 : 1445 – 1452 .
- Xiao , Y. , Song , M. H. and Liu , M. Z. 2011 . Convergence and stability of the semi-implicit Euler method with variable step size for a linear stochastic pantograph differential equation . Int. J. Numer. Anal. Model. , 2 : 214 – 225 .