Abstract and Applied Analysis

Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps

Abstract

This paper is concerned with the stability of analytical and numerical solutions for nonlinear stochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean-square exponential stability of the exact solution is derived. Then, mean-square stability of the numerical solution is investigated. It is shown that the compensated stochastic θ methods inherit stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsize $\Delta t=\tau /m$ when $1/2\le \theta \le 1$, and they are exponentially mean-square stable if the stepsize $\Delta t\in (0,\Delta {t}_{0})$ when $0\le \theta <1$. Finally, some numerical experiments are given to illustrate the theoretical results.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 831082, 13 pages.

Dates
First available in Project Euclid: 1 April 2013

https://projecteuclid.org/euclid.aaa/1364845178

Digital Object Identifier
doi:10.1155/2012/831082

Mathematical Reviews number (MathSciNet)
MR2889088

Zentralblatt MATH identifier
1236.60055

Citation

Li, Qiyong; Gan, Siqing. Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 831082, 13 pages. doi:10.1155/2012/831082. https://projecteuclid.org/euclid.aaa/1364845178

References

• R. Cont and P. Tankov, Financial Modelling with Jump Processes, Financial Mathematics Series, Chapman & Hall, Boca Raton, Fla, USA, 2004.
• 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.
• K. Sobczyk, Stochastic Differential Equations. With Applications to Physics and Engineerin, Kluwer Academic, Dordrecht, The Netherlands, 1991.
• P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, Germany, 1992.
• G. N. Milstein and M. V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer, Berlin, Germany, 2004.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, vol. 64, Springer, Berlin, Germany, 2010.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, UK, 1997.
• 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.
• E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic Problem, Springer, Berlin, Germany, 2nd edition, 1996.
• 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.