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Convergence rate of EM scheme for SDDEs


Authors: Jianhai Bao and Chenggui Yuan
Journal: Proc. Amer. Math. Soc. 141 (2013), 3231-3243
MSC (2010): Primary 65C30; Secondary 60H10, 65L20
DOI: https://doi.org/10.1090/S0002-9939-2013-11886-1
Published electronically: May 13, 2013
MathSciNet review: 3068976
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Abstract: In this paper we investigate the convergence rate of the Euler-Maruyama (EM) scheme for a class of stochastic differential delay equations, where the corresponding coefficients may be highly nonlinear with respect to the delay variables. In particular, we reveal that the convergence rate of the Euler-Maruyama scheme is $ \frac {1}{2}$ for the Brownian motion case, while we show that it is best to use the mean-square convergence for the pure-jump case and that the order of mean-square convergence is close to $ \frac {1}{2}$.


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  • 1. J. Bao, B. Böttcher, X. Mao and C. Yuan, Convergence rate of numerical solutions to SFDEs with jumps, J. Comput. Appl. Math., 236 (2011), no. 2, 119-131. MR 2827394
  • 2. I. Gyöngy and M. Rásonyi, A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients, Stochastic Process. Appl., 121 (2011), no. 10, 2189-2200. MR 2822773
  • 3. I. Gyöngy, A note on Euler's approximations, Potential Anal., 8 (1998), no. 3, 205-216. MR 1625576 (99d:60060)
  • 4. D. J. Higham and P. E. Kloeden, Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math., 101 (2005), no. 1, 101-119. MR 2194720 (2006k:65013)
  • 5. D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), no. 3, 1041-1063. MR 1949404 (2003j:65005)
  • 6. Hu, Y., Semi-implicit Euler-Maruyama scheme for stiff stochastic equations, The Silivri Workshop, Progr. Probab. 38, H. Koerezlioglu, ed., Birkhäuser, Boston, 1996, 183-202. MR 1396331 (97e:60101)
  • 7. N. Jacob, Y. Wang and C. Yuan, Numerical solutions of stochastic differential delay equations with jumps, Stoch. Anal. Appl., 27 (2009), no. 4, 825-853. MR 2541379 (2010j:60144)
  • 8. P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. MR 1214374 (94b:60069)
  • 9. U. Küchler and E. Platen, Strong discrete time approximation of stochastic differential equations with time delay, Math. Comput. Simulation, 54 (2000), no. 1-3, 189-205. MR 1800113 (2001j:60111)
  • 10. X. Mao and S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, J. Comput. Appl. Math., 151 (2003), no. 1, 215-227. MR 1950237 (2003k:65008)
  • 11. C. Marinelli, C. Prévôt and M. Röckner, Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise, J. Funct. Anal., 258 (2010), no. 2, 616-649. MR 2557949 (2011a:60230)
  • 12. E. Platen and N. Bruti-Liberati, Numerical solution of stochastic differential equations with jumps in finance, Springer-Verlag, Berlin, 2010. MR 2723480 (2012b:60001)
  • 13. H. Schurz, Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications, Logos Verlag Berlin, Berlin, 1997. MR 1991701
  • 14. F. Wu, X. Mao and K. Chen, The Cox-Ingersoll-Ross model with delay and strong convergence of its Euler-Maruyama approximate solutions, Appl. Numer. Math., 59 (2009), no. 10, 2641-2658. MR 2553159 (2011a:91290)
  • 15. C. Yuan and X. Mao, A note on the rate of convergence of the Euler-Maruyama method for stochastic differential equations, Stoch. Anal. Appl., 26 (2008), no. 2, 325-333. MR 2399739 (2009b:65013)

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Additional Information

Jianhai Bao
Affiliation: Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, United Kingdom
Email: majb@swansea.ac.uk

Chenggui Yuan
Affiliation: Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, United Kingdom
Email: C.Yuan@swansea.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2013-11886-1
Keywords: Stochastic differential delay equation, highly nonlinear, jumps, EM scheme, convergence rate
Received by editor(s): November 17, 2011
Published electronically: May 13, 2013
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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