Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-13T03:37:28.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Strong Convergence Analysis of Split-Step θ-Scheme for Nonlinear Stochastic Differential Equations with Jumps

Part of: Stochastic analysis Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  19 September 2016

Xu Yang  and
Weidong Zhao
Xu Yang*
Affiliation:
School of Mathematics, Shandong University, Jinan 250100, China
Weidong Zhao*
Affiliation:
School of Mathematics, Shandong University, Jinan 250100, China
*
*Corresponding author. Email:dshyangxu@163.com (X. Yang), wdzhao@sdu.edu.cn (W. D. Zhao)
*Corresponding author. Email:dshyangxu@163.com (X. Yang), wdzhao@sdu.edu.cn (W. D. Zhao)
Get access

Abstract

In this paper, we investigate the mean-square convergence of the split-step θ-scheme for nonlinear stochastic differential equations with jumps. Under some standard assumptions, we rigorously prove that the strong rate of convergence of the split-step θ-scheme in strong sense is one half. Some numerical experiments are carried out to assert our theoretical result.

Keywords

Split-step schemestrong convergencestochastic differential equationjump-diffusionone-side Lipschitz condition

MSC classification

Secondary: 65C20: Models, numerical methods 60H35: Computational methods for stochastic equations 60H10: Stochastic ordinary differential equations
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 6 , December 2016 , pp. 1004 - 1022
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bruti-Liberati, N. and Platen, E., Strong approximations of stochastic differential equations with jumps, J. Comput. Appl. Math., 205 (2007), pp. 982–1001.CrossRefGoogle Scholar
[2] Chalmers, G. and Higham, D., Asymptotic stability of a jump-diffusion equation and its numerical approximation, SIAM J. Sci. Comput., 31 (2008), pp. 1141–1155.CrossRefGoogle Scholar
[3] Chalmers, G. and Higham, D., First and second moment reversion for a discretized square root process with jumps, J. Differ. Equ. Appl., 16 (2010), pp. 143–156.CrossRefGoogle Scholar
[4] Cont, R. and Tankov, P., Financial Modelling with Jump Processes, Financial Mathematics Series, Chapman & Hall/CRC, London, Boca Raton, 2004.Google Scholar
[5] Gikhman, I. I. and Skorokhod, A. V., Stochastic Differential Equations, Springer-Verlag, Berlin, 1972.CrossRefGoogle Scholar
[6] Ding, X., Ma, Q. and L.|Zhang, Convergence and stability of the spilit-step θ-method for stochastic differential equations, Comput. Math. Appl., 60 (2010), pp. 1310–1321.CrossRefGoogle Scholar
[7] Glasserman, P. and Merener, N., Numerical solution of jump-diffusin LIBOR market models, Financ. Stoch., 7 (2003), pp. 1–27.CrossRefGoogle Scholar
[8] Higham, D., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43(3) (2002), pp. 525–546.CrossRefGoogle Scholar
[9] Higham, D. and Mao, X., Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), pp. 1041–1063.CrossRefGoogle Scholar
[10] Higham, D. and Kloeden, P., Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math., 101 (2005), pp. 101–119.CrossRefGoogle Scholar
[11] Higham, D. and Kloeden, P., Convergence and stability of implicit methods for jump-diffusion systems, Int. J. Numer. Anal. Model., 3 (2006), pp. 125–140.Google Scholar
[12] Higham, D. and Kloeden, P., Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems, J. Comput. Appl. Math., 205 (2007), pp. 949–956.CrossRefGoogle Scholar
[13] Huang, C., Exponential mean square stability of numerical methods for systems of stochastic differential equations, J. Comput. Appl. Math., 236 (2012), pp. 4016–4026.CrossRefGoogle Scholar
[14] Hutzenthler, M., Jentzen, A. and Kloeden, P., Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz coefficients, Proc. R. Soc. Lond. A Nsath. Phys. Eng. Sci., 467 (2011), pp. 1563–1576.Google Scholar
[15] Kahl, C. and Schurz, H., Balanced Milstein methods for ordinary SDEs, Monte Carlo Methods Appl., 12(2) (2006), pp. 143–170.CrossRefGoogle Scholar
[16] Karatzas, I. and Shreve, S. E., Brownian Motion and Stochastic Calculus, 2nd ed., Springer-Verlag, Berlin, 1991.Google Scholar
[17] Kloeden, P. and Platen, E., Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992.CrossRefGoogle Scholar
[18] Maghsoodi, Y., Mean square efficient numerical solution of jump-diffusion stochastic differential equations, Indian J. Statistics, 58 (1996), pp. 25–47.Google Scholar
[19] Maghsoodi, Y., Exact solutions and doubly efficient approximations and simulation of jump-diffsion Ito equations, Stochatic Anal. Appl., 16 (1998), pp. 1049–1072.CrossRefGoogle Scholar
[20] Mao, X., Stochastic Differential Equations and Applications, Horwood, New York, 1997.Google Scholar
[21] Milstein, G. N., Platen, E. and Schurz, H., Balanced implicit methods for stiff stochastic systems, SIAM J. Numer. Anal., 35(3) (1998), pp. 1010–1019.CrossRefGoogle Scholar
[22] Wang, X. and Gan, S., B-convergence of split-step one-leg theta methods for stochastic differential equations, J. Appl. Math. Comput., 38 (2012), pp. 489–503.CrossRefGoogle Scholar
[23] Wang, X. and Gan, S., Compensated stochastic theta methods for stochastic differential equations with jumps, Appl. Numer. Math., 60 (2012), pp. 877–887.CrossRefGoogle Scholar
[24] Wu, F., Mao, X. and Chen, K., Strong convergence of Monte Carlo simulations of the mean-reverting square root process with jump, Appl. Math. Comput., 206(2) (2008), pp. 494–505.Google Scholar
[25] Zhao, W., Tian, L. and Ju, L., Convergence analysis of a splitting scheme for stochastic differential equations, Int. J. Numer. Anal. Mod., 4 (2008), pp. 673–692.Google Scholar