Exponential mean-square stability of the θ-method for neutral stochastic delay differential equations with jumps

References

• Cao, W., Hao, P., & Zhang, Z. (2014). Split-step θ-method for stochastic delay differential equations. Applied Numerical Mathematics, 76, 19–33.
   Web of Science ®Google Scholar
• Cao, W., Liu, M., & Fan, Z. (2004). MS-stability of the Euler–Maruyama method for stochastic differential delay equations. Applied Mathematics and Computation, 159(1), 127–135.
   Web of Science ®Google Scholar
• Chen, L., & Wu, F. (2012). Almost sure exponential stability of the θ-method for stochastic differential equations. Statistics and Probability Letters, 82(9), 1669–1676.
   Web of Science ®Google Scholar
• Gan, S., Schurz, H., & Zhang, H. (2011). Mean-square convergence of stochastic θ-methods for nonlinear neutral stochastic delay differential equations. International Journal of Numerical Analysis and Modeling, 8(2), 201–213.
   Web of Science ®Google Scholar
• Grigoriu, M. (2004). Dynamic systems with Poisson white noise. Nonlinear Dynamics, 36(2–4), 255–266.
   Web of Science ®Google Scholar
• Higham, D.J. (2000). Mean-square and asymptotic stability of the stochastic theta method. SIAM Journal on Numerical Analysis, 38(3), 753–769.
   Web of Science ®Google Scholar
• Higham, D.J., & Kloeden, P.E. (2005). Numerical methods for nonlinear stochastic differential equations with jumps. Numerische Mathematik, 101(1), 101–119.
   Web of Science ®Google Scholar
• Higham, D.J., Mao, X., & Yuan, C. (2007). Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. SIAM Journal on Numerical Analysis, 45(2), 592–609.
   Web of Science ®Google Scholar
• Hu, L. (2012). Convergence and stability of numerical methods for several classes of stochastic differential equations with Poisson-driven jumps [master's thesis] (in Chinese). Central South University, Changsha, China.
   Google Scholar
• Hu, L., & Gan, S. (2011). Mean-square convergence of drift-implicit one-step methods for neutral stochastic delay differential equations with jump diffusion. Discrete Dynamics in Nature and Society. Advance online publication. doi:10.1155/2011/917892
   Web of Science ®Google Scholar
• Huang, C. (2014). Mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations. Journal of Computational and Applied Mathematics, 259, 77–86.
   Web of Science ®Google Scholar
• Jacob, N., Wang, Y., & Yuan, C. (2009). Stochastic differential delay equations with jumps, under nonlinear growth condition. Stochastics: An International Journal of Probability and Stochastics Processes, 81(6), 571–588.
   Web of Science ®Google Scholar
• Kloeden, P.E., & Platen, E. (1992). Numerical solution of stochastic differential equations. Berlin: Springer-Verlag.
   Google Scholar
• Li, X., & Cao, W. (2015). On mean-square stability of two-step Maruyama methods for nonlinear neutral stochastic delay differential equations. Applied Mathematics and Computation, 261, 373–381.
   Web of Science ®Google Scholar
• Li, Q., & Gan, S. (2011). Almost sure exponential stability of numerical solutions for stochastic delay differential equations with jumps. Journal of Applied Mathematics and Computing, 37(1–2), 541–557.
   Google Scholar
• Li, Q., & Gan, S. (2012). Mean-square exponential stability of stochastic theta methods for nonlinear stochastic delay integro-differential equations. Journal of Applied Mathematics and Computing, 39(1–2), 69–87.
   Google Scholar
• Li, Q., Gan, S., & Wang, X. (2013). Compensated stochastic theta methods for stochastic differential delay equations with jumps. International Journal of Computer Mathematics, 90(5), 1057–1071.
   Web of Science ®Google Scholar
• Liu, D., Yang, G., & Zhang, W. (2011). The stability of neutral stochastic delay differential equations with Poisson jumps by fixed points. Journal of Computational and Applied Mathematics, 235(10), 3115–3120.
   Web of Science ®Google Scholar
• Mao, X. (1997). Stochastic differential equations and applications. Chichester: Horwood.
   Google Scholar
• Mao, X. (2000). Asymptotic properties of neutral stochastic differential delay equations. Stochastics: An International Journal of Probability and Stochastic Processes, 68(3–4), 273–295.
   Google Scholar
• Mariton, M. (1990). Jump linear systems in automatic control. New York, NY: Marcel Dekker.
   Google Scholar
• Ren, Y., & Xia, N. (2009). Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay. Applied Mathematics and Computation, 210(1), 72–79.
   Web of Science ®Google Scholar
• Song, B., Park, J.H., Wu, Z., & Zhang, Y. (2013). New results on delay-dependent stability analysis for neutral stochastic delay systems. Journal of the Franklin Institute, 350(4), 840–852.
   Web of Science ®Google Scholar
• Tan, J., & Wang, H. (2011). Mean-square stability of the Euler–Maruyama method for stochastic differential delay equations with jumps. International Journal of Computer Mathematics, 88(2), 421–429.
   Web of Science ®Google Scholar
• Tan, J., Wang, H., & Guo, Y. (2012). Existence and uniqueness of solutions to neutral stochastic functional differential equations with Poisson jumps. Abstract and Applied Analysis. Advance online publication. doi:10.1155/2012/371239
   Google Scholar
• Tan, J., Wang, H., Guo, Y., & Zhu, Z. (2014). Numerical solutions to neutral stochastic delay differential equations with Poisson jumps under local Lipschitz condition. Mathematical Problems in Engineering.Advance online publication. doi:10.1155/2014/976183
   Web of Science ®Google Scholar
• Wang, W. (2011). Mean-square stability of Euler method for nonlinear neutral stochastic delay differential equations. 2011 IEEE 2nd international conference on Computing, Control and Industrial Engineering (CCIE) (Vol. 2, pp. 366–369). Hubei, Wuha.
   Google Scholar
• Wang, W., & Chen, Y. (2011). Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations. Applied Numerical Mathematics, 61(5), 696–701.
   Web of Science ®Google Scholar
• Wang, L., Mei, C., & Xue, H. (2007). The semi-implicit Euler method for stochastic differential delay equation with jumps. Applied Mathematics and Computation, 192(2), 567–578.
   Web of Science ®Google Scholar
• Wu, F., Mao, X., & Szpruch, L. (2010). Almost sure exponential stability of numerical solutions for stochastic delay differential equations. Numerische Mathematik, 115(4), 681–697.
   Web of Science ®Google Scholar
• Yu, Z. (2015). Almost sure and mean square exponential stability of numerical solutions for neutral stochastic functional differential equations. International Journal of Computer Mathematics, 92(1), 132–150.
   Web of Science ®Google Scholar
• Zhou, S., & Wu, F. (2009). Convergence of numerical solutions to neutral stochastic delay differential equations with Markovian switching. Journal of Computational and Applied Mathematics, 229(1), 85–96.
   Web of Science ®Google Scholar
• Zong, X., & Wu, F. (2014). Choice of θ and mean-square exponential stability in the stochastic theta method of stochastic differential equations. Journal of Computational and Applied Mathematics, 255, 837–847.
   Web of Science ®Google Scholar
• Zong, X., Wu, F., & Huang, C. (2015). Exponential mean square stability of the theta approximations for neutral stochastic differential delay equations. Journal of Computational and Applied Mathematics, 286, 172–185.
   Web of Science ®Google Scholar