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Approximation of Euler–Maruyama for one-dimensional stochastic differential equations involving the maximum process

  • Kamal Hiderah ORCID logo EMAIL logo

Abstract

The aim of this paper is to show the approximation of Euler–Maruyama Xtn for one-dimensional stochastic differential equations involving the maximum process. In addition to that it proves the strong convergence of the Euler–Maruyama whose both drift and diffusion coefficients are Lipschitz. After that, it generalizes to the non-Lipschitz case.

Acknowledgements

The author would like to thank the referees and the editor for the very useful comments and suggestions, which have helped to improve the paper a lot. The author is very grateful to Professor Mohsine Benabdallah for suggesting this problem, and for fruitful discussions.

References

[1] R. Avikainen, On irregular functionals of SDEs and the Euler scheme, Finance Stoch. 13 (2009), no. 3, 381–401. 10.1007/s00780-009-0099-7Search in Google Scholar

[2] V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function, Probab. Theory Related Fields 104 (1996), no. 1, 43–60. 10.1007/BF01303802Search in Google Scholar

[3] R. Belfadli, S. Hamadène and Y. Ouknine, On one-dimensional stochastic differential equations involving the maximum process, Stoch. Dyn. 9 (2009), no. 2, 277–292. 10.1142/S0219493709002671Search in Google Scholar

[4] K. S. Chan and O. Stramer, Weak consistency of the Euler method for numerically solving stochastic differential equations with discontinuous coefficients, Stochastic Process. Appl. 76 (1998), no. 1, 33–44. 10.1016/S0304-4149(98)00020-9Search in Google Scholar

[5] L. Chaumont and R. A. Doney, Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion, Probab. Theory Related Fields 113 (1999), no. 4, 519–534. 10.1007/s004400050216Search in Google Scholar

[6] L. Chaumont and R. A. Doney, Some calculations for doubly perturbed Brownian motion, Stochastic Process. Appl. 85 (2000), no. 1, 61–74. 10.1016/S0304-4149(99)00065-4Search in Google Scholar

[7] L. Chaumont, R. A. Doney and Y. Hu, Upper and lower limits of doubly perturbed Brownian motion, Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 2, 219–249. 10.1016/S0246-0203(00)00123-0Search in Google Scholar

[8] B. Davis, Weak limits of perturbed random walks and the equation Yt=Bt+αsup{Ys:st}+βinf{Ys:st}, Ann. Probab. 24 (1996), no. 4, 2007–2023. Search in Google Scholar

[9] 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. 10.1016/j.spa.2011.06.008Search in Google Scholar

[10] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43 (2001), no. 3, 525–546. 10.1137/S0036144500378302Search in Google Scholar

[11] 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. 10.1137/S0036142901389530Search in Google Scholar

[12] K. Ito, On stochastic differential equations, Mem. Amer. Math. Soc. No. 4 (1951), 1–57. 10.1007/978-1-4612-5370-9_10Search in Google Scholar

[13] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Appl. Math. (New York) 23, Springer, Berlin, 1992. 10.1007/978-3-662-12616-5Search in Google Scholar

[14] A. Kohatsu-Higa, A. Lejay and K. Yasuda, On weak approximation of stochastic differential equations with discontinuous drift coeffcient, preprint (2012), https://hal.inria.fr/hal-00670123/document. Search in Google Scholar

[15] H.-L. Ngo and D. Taguchi, Strong rate of convergence for the Euler–Maruyama approximation of stochastic differential equations with irregular coefficients, Math. Comp. 85 (2016), no. 300, 1793–1819. 10.1090/mcom3042Search in Google Scholar

[16] M. Perman and W. Werner, Perturbed Brownian motions, Probab. Theory Related Fields 108 (1997), no. 3, 357–383. 10.1007/s004400050113Search in Google Scholar

[17] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971), 155–167. 10.1215/kjm/1250523691Search in Google Scholar

[18] L. Yan, The Euler scheme with irregular coefficients, Ann. Probab. 30 (2002), no. 3, 1172–1194. 10.1214/aop/1029867124Search in Google Scholar

Received: 2019-08-30
Accepted: 2020-01-25
Published Online: 2020-02-11
Published in Print: 2020-03-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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