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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:[email protected] (X. Yang), [email protected] (W. D. Zhao)
*Corresponding author. Email:[email protected] (X. Yang), [email protected] (W. D. Zhao)
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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 

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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