References
- Baker , C. T.H. and Buckwar , E. 2000 . Introduction to the numerical analysis of stochastic delay differential equations . J. Comput. Appl. Math. , 125 : 297 – 307 . (doi:10.1016/S0377-0427(00)00475-1)
- Baker , C. T.H. and Buckwar , E. 2005 . Exponential stability in p-th mean of solutions, and of convergent Euler-stype solutions, of stochastic delay differential equations . J. Comput. Appl. Math. , 184 : 404 – 427 . (doi:10.1016/j.cam.2005.01.018)
- Bellen , A. and Zennaro , M. 2003 . Numerical Methods for Delay Differential Equations , Oxford : Oxford University Press .
- Buckwar , E. and Riedler , M. 2011 . Runge-Kutta methods for jump-diffusion differential equations . J. Comput. Appl. Math. , 236 : 1155 – 1182 . (doi:10.1016/j.cam.2011.08.001)
- Burrage , K. , Burrage , P. M. and Tian , T. 2004 . Numerical methods for strong solutions of stochastic differential equations: An overview . Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 460 : 373 – 402 . (doi:10.1098/rspa.2003.1247)
- Calvo , M. and Grande , T. 1988 . On the asymptotic stability of θ-methods for delay differential equations . Numer. Math. , 54 : 257 – 269 . (doi:10.1007/BF01396761)
- Cont , R. and Tankov , P. 2003 . Financial Modelling with Jump Processes , London : Chapman and Hall/CRC Press .
- Hairer , E. and Wanner , G. 1996 . Solving Ordinary Differential Equations II: Stiff and Differential- Algebraic Problems , 2 , Berlin : Springer-Verlag .
- Higham , D. J. 2000 . Mean-square and asymptotic stability of the stochastic theta method . SIAM J. Numer. Anal. , 38 : 753 – 769 . (doi:10.1137/S003614299834736X)
- Higham , D. J. and Kloeden , P. E. 2005 . Numerical methods for nonlinear stochastic differential equations with jumps . Numer. Math. , 101 : 101 – 119 . (doi:10.1007/s00211-005-0611-8)
- Higham , D. J. and Kloeden , P. E. 2006 . Convergence and stability of implicit methods for jump-diffusion systems . Int. J. Numer. Anal. Model. , 3 : 125 – 140 .
- Hu , P. and Huang , C. 2011 . Stability of stochastic θ-methods for stochastic delay integro-differential equations . Int. J. Comput. Math. , 88 : 1417 – 1429 . (doi:10.1080/00207160.2010.509430)
- Jiang , F. , Shen , Y. and Liu , L. 2011 . Taylor approximation of the solutions of stochastic differential delay equations with Poisson jump . Commun. Nonlinear. Sci. Numer. Simul. , 16 : 798 – 804 . (doi:10.1016/j.cnsns.2010.04.032)
- Kloeden , P. E. and Platen , E. 1992 . Numerical Solution of Stochastic Differential Equations , Berlin : Springer .
- Küchler , U. and Platen , E. 2000 . Strong discrete time approximation of stochastic differential equations with time delay . Math. Comput. Simulation , 54 : 189 – 205 . (doi:10.1016/S0378-4754(00)00224-X)
- Li , R. and Chang , Z. 2007 . Convergence of numerical solution to stochastic delay differential equation with Poisson jump and Markovian switching . Appl. Math. Comput. , 184 : 451 – 463 . (doi:10.1016/j.amc.2006.06.112)
- Li , Q. and Gan , S. 2012 . Stability of analytical and numerical solutions for nonlinear stochastic delay differential equations with jumps . Abstr. Appl. Anal. , Article ID 831082, 13 pages
- Liberati , N. and Platen , E. 2007 . Approximation of jump diffusions in finance and economics . Comput. Econom. , 29 : 283 – 312 . (doi:10.1007/s10614-006-9066-y)
- Liu , M. , Cao , W. and Fan , Z. 2004 . Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation . J. Comput. Appl. Math. , 170 : 255 – 268 . (doi:10.1016/j.cam.2004.01.040)
- Mao , X. 1997 . Stochastic Differential Equations and Applications , New York : Horwood .
- Milstein , G. N. and Tretyakov , M. V. 2004 . Stochastic Numerics for Mathematical Physics , Berlin : Springer .
- Mohamad , S. and Gopalsamy , K. 2000 . Continuous and discrete Halanay-type inequalitites . Bull. Austral. Math. Soc. , 61 : 371 – 385 . (doi:10.1017/S0004972700022413)
- Sobczyk , K. 1991 . Stochastic Differential Equations: With Application to Physics and Engineering , Dordrecht : Kluwer Academic .
- Tan , J. and Wang , H. 2011 . Mean-square stability of the Euler-Maruyama method for stochastic differential delay equations with jumps . Int. J. Comput. Math. , 88 : 421 – 429 .
- Wang , X. and Gan , S. 2010 . Compensated stochastic theta methods for stochastic differential equations with jumps . Appl. Num. Math. , 60 : 877 – 887 . (doi:10.1016/j.apnum.2010.04.012)
- Wang , L. , Mei , C. and Xue , H. 2007 . The semi-implicit Euler method for stochastic differential delay equation with jumps . Appl. Math. Comput. , 192 : 567 – 578 . (doi:10.1016/j.amc.2007.03.027)
- X. Wang, S. Gan, and D. Wang, θ-Maruyama methods for nonlinear stochastic differential delay equations, submitted.
- Wu , F. , Mao , X. and Szpruch , L. 2010 . Almost sure exponential stability of numerical solutions for stochastic delay differential equations . Numer. Math. , 115 : 681 – 697 . (doi:10.1007/s00211-010-0294-7)