Li, Qiyong; Gan, Siqing Stability of analytical and numerical solutions for nonlinear stochastic delay differential equations with jumps. (English) Zbl 1236.60055 Abstr. Appl. Anal. 2012, Article ID 831082, 13 p. (2012). Summary: This paper is concerned with the stability of analytical and numerical solutions for nonlinear stochastic delay differential equations with jumps. A sufficient condition for the mean-square exponential stability of the exact solution is derived. Then, the mean-square stability of the numerical solution is investigated. It is shown that the compensated stochastic \(\theta\) methods inherit the stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsize \(\Delta t = \tau/m\) when \(1/2 \leq \theta \leq 1\), and they are exponentially mean-square stable if the stepsize \(\Delta t \in (0, \Delta t_0)\) when \(0 \leq \theta < 1\). Finally, some numerical experiments are given to illustrate the theoretical results. Cited in 2 Documents MSC: 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 65C30 Numerical solutions to stochastic differential and integral equations 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) Keywords:analytical and numerical solutions; nonlinear stochastic delay differential equations; mean-square stability Software:RODAS PDFBibTeX XMLCite \textit{Q. Li} and \textit{S. Gan}, Abstr. Appl. Anal. 2012, Article ID 831082, 13 p. (2012; Zbl 1236.60055) Full Text: DOI References: [1] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Financial Mathematics Series, Chapman & Hall, Boca Raton, Fla, USA, 2004. · Zbl 1206.82018 [2] N. Bruti-Liberati and E. Platen, “Approximation of jump diffusions in finance and economics,” Computational Economics, vol. 29, no. 3-4, pp. 283-312, 2007. · Zbl 1161.91384 [3] K. Sobczyk, Stochastic Differential Equations. With Applications to Physics and Engineerin, Kluwer Academic, Dordrecht, The Netherlands, 1991. · Zbl 1020.81884 [4] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, Germany, 1992. · Zbl 0925.65261 [5] G. N. Milstein and M. V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer, Berlin, Germany, 2004. · Zbl 1121.91367 [6] C. T. H. Baker and E. Buckwar, “Introduction to the numerical analysis of stochastic delay differential equations,” Journal of Computational and Applied Mathematics, vol. 125, no. 1-2, pp. 297-307, 2000. · Zbl 0971.65004 [7] M. Liu, W. Cao, and Z. Fan, “Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation,” Journal of Computational and Applied Mathematics, vol. 170, no. 2, pp. 255-268, 2004. · Zbl 1059.65006 [8] H. Zhang, S. Gan, and L. Hu, “The split-step backward Euler method for linear stochastic delay differential equations,” Journal of Computational and Applied Mathematics, vol. 225, no. 2, pp. 558-568, 2009. · Zbl 1183.65007 [9] X. Ding, K. Wu, and M. Liu, “Convergence and stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations,” International Journal of Computer Mathematics, vol. 83, no. 10, pp. 753-763, 2006. · Zbl 1115.65007 [10] P. Hu and C. Huang, “Stability of stochastic \theta -methods for stochastic delay integro-differential equations,” International Journal of Computer Mathematics, vol. 88, no. 7, pp. 1417-1429, 2011. · Zbl 1222.65010 [11] W. Wang and Y. Chen, “Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations,” Applied Numerical Mathematics, vol. 61, no. 5, pp. 696-701, 2011. · Zbl 1210.65014 [12] D. J. Higham and P. E. Kloeden, “Numerical methods for nonlinear stochastic differential equations with jumps,” Numerische Mathematik, vol. 101, no. 1, pp. 101-119, 2005. · Zbl 1186.65010 [13] D. J. Higham and P. E. Kloeden, “Convergence and stability of implicit methods for jump-diffusion systems,” International Journal of Numerical Analysis and Modeling, vol. 3, no. 2, pp. 125-140, 2006. · Zbl 1109.65007 [14] E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, vol. 64, Springer, Berlin, Germany, 2010. · Zbl 1225.60004 [15] X. Wang and S. Gan, “Compensated stochastic theta methods for stochastic differential equations with jumps,” Applied Numerical Mathematics, vol. 60, no. 9, pp. 877-887, 2010. · Zbl 1198.65034 [16] L. Ronghua and C. Zhaoguang, “Convergence of numerical solution to stochastic delay differential equation with Poisson jump and Markovian switching,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 451-463, 2007. · Zbl 1120.65003 [17] L.-s. Wang, C. Mei, and H. Xue, “The semi-implicit Euler method for stochastic differential delay equations with jumps,” Applied Mathematics and Computation, vol. 192, no. 2, pp. 567-578, 2007. · Zbl 1193.65008 [18] N. Jacob, Y. Wang, and C. Yuan, “Numerical solutions of stochastic differential delay equations with jumps,” Stochastic Analysis and Applications, vol. 27, no. 4, pp. 825-853, 2009. · Zbl 1168.60356 [19] D. Liu, “Mean square stability of impulsive stochastic delay differential equations with Markovian switching and Poisson jumps,” International Journal of Computational and Mathematical Sciences, vol. 5, no. 1, pp. 58-61, 2011. [20] J. Tan and H. Wang, “Mean-square stability of the Euler-Maruyama method for stochastic differential delay equations with jumps,” International Journal of Computer Mathematics, vol. 88, no. 2, pp. 421-429, 2011. · Zbl 1215.65016 [21] X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, UK, 1997. · Zbl 0892.60057 [22] S. Mohamad and K. Gopalsamy, “Continuous and discrete Halanay-type inequalities,” Bulletin of the Australian Mathematical Society, vol. 61, no. 3, pp. 371-385, 2000. · Zbl 0958.34008 [23] E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic Problem, Springer, Berlin, Germany, 2nd edition, 1996. · Zbl 0859.65067 [24] C. T. H. Baker and E. Buckwar, “On Halanay-type analysis of exponential stability for the \theta -Maruyama method for stochastic delay differential equations,” Stochastics and Dynamics, vol. 5, no. 2, pp. 201-209, 2005. · Zbl 1081.65012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.