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Section B

Almost sure and mean square exponential stability of numerical solutions for neutral stochastic functional differential equations

Pages 132-150 | Received 19 Dec 2012, Accepted 13 Jan 2014, Published online: 27 Mar 2014

References

  • S. Gan, H. Schurz, and H. Zhang, Mean square convergence of stochastic θ-methods for nonlinear neutral stochastic differential delay equations, Inter. J. Numer. Anal. Model. 8 (2011), pp. 201–213.
  • L. Huang and F. Deng, Razumikhin-type theorems on stability of neutral stochastic functional differential equations, IEEE Trans. Automat. Control 53(7) (2008), pp. 1718–1723. doi: 10.1109/TAC.2008.929383
  • S. Jankovic, J. Randjelovic, and M. Jovanovic, Razumikhin-type exponential stability criteria of neutral stochastic functional differential equations, J. Math. Anal. Appl. 355 (2009), pp. 811–820. doi: 10.1016/j.jmaa.2009.02.011
  • R. Lipster and A. Shiryayev, Title of a Book, Theory of Martingale, Horwood, Chichester, 1989.
  • Q. Li and S. Gan, Almost sure exponential stability of numerical solutions for stochastic delay differential equations with jumps, J. Appl. Math. Comput. 37 (2011), pp. 541–557. doi: 10.1007/s12190-010-0449-9
  • X. Mao, Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. Anal. 28(2) (1997), pp. 389–401. doi: 10.1137/S0036141095290835
  • M. Milosevic, Highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama methods, Math. Comput. Model. 54 (2011), pp. 2235–2251. doi: 10.1016/j.mcm.2011.05.033
  • M. Milosevic, Almost sure exponential stability of solutions to highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama approximation, Math. Comput. Model. 57 (2013), pp. 887–899. doi: 10.1016/j.mcm.2012.09.016
  • A. Rodkina and H. Schurz, Almost sure asymptotic stability of drift-implicit θ-methods for bilinear ordinary stochastic differential equations in ℝ1, J. Comput. Appl. Math. 180 (2005), pp. 13–31. doi: 10.1016/j.cam.2004.09.060
  • L. Shaikkhet, Some new aspects of Lyapunov-type theorems for stochastic differential equations of neutral type, SIAM J. Control Optim. 48 (2010), pp. 4481–4499. doi: 10.1137/080744165
  • A.N. Shiryayev, Title of a Book, Probability, Springer, Berlin, 1996.
  • F. Wu and S. Hu, Razumikhin-type theorem for neutral stochastic functional differential equations with unbounded delay, Acta Math. Sci. 31B(4) (2011), pp. 1245–1258.
  • F. Wu and X. Mao, Numerical solutions of neutral stochastic functional differential equations, SIAM J. Numer. Anal. 46(4) (2008), pp. 1821–1841. doi: 10.1137/070697021
  • F. Wu, X. Mao, and L. Szpruch, Almost sure exponential stability of numerical solutions for stochastic delay differential equations, Numer. Math. 115 (2010), pp. 681–697. doi: 10.1007/s00211-010-0294-7
  • F. Wu, X. Mao, and P. Kloeden, Almost sure exponential stability of Euler-Maruyama approximations for stochastic functional differential equations, Random Oper. Stoch. Equ. 19 (2011), pp.165–186. doi: 10.1515/ROSE.2011.010
  • W. Wang, Y. Chen, Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations, Appl. Numer. Math. 61 (2011), pp. 696–701. doi: 10.1016/j.apnum.2011.01.003
  • M. Xue, S. Zhou, and S. Hu, Stability of nonlinear neutral stochastic functional differential equations, J. Appl. Math. 2010 (2010), article id: 425762, 26 pp. 10.1155/2010/425762
  • B. Yin and Z. Ma, Convergence of the semi-implicit Euler method for neutral stochastic delay differential equations with phase semi-Markovian switching, Appl. Math. Model. 35 (2011), pp. 2094–2109. doi: 10.1016/j.apm.2010.11.002
  • Z. Yu and M. Liu, Almost surely asymptotic stability of numerical solutions for neutral stochastic delay differential equations, Discrete Dyn. Nat. Soc. 2011 (2011), article id: 217672, 11 pp. 10.1155/2011/217672
  • Z. Yu, Almost surely asymptotic stability of exact and numerical solutions for neutral stochastic pantograph equations, Abst. Appl. Anal. 2011 (2011), article id: 143079, 14 pp. 10.1155/2011/143079
  • Z. Yu, The improved stability analysis of the backward Euler method for neutral stochastic delay differential equations, Int. J. Comput. Math. 90 (2013), pp. 1489–1494. doi: 10.1080/00207160.2012.756479
  • S. Zhou and S. Hu, Razumikhin-type theorems of neutral stochastic functional differential equations, Acta Math. Sci. 29B(1) (2009), pp. 181–190.
  • S. Zhou and F. Wu, Convergence of numerical solutions to neutral stochastic delay differential equations with Markovian switching, J. Comput. Appl. Math. 229 (2009), pp. 85–96. doi: 10.1016/j.cam.2008.10.013
  • S. Zhou and Z. Fang, Numerical approximation of nonlinear neutral stochastic functional differential equations, J. Appl. Math. Comput. 41 (2013), pp. 427–445. doi: 10.1007/s12190-012-0605-5

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