[go: up one dir, main page]

Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter October 9, 2018

Strong rate of convergence for the Euler–Maruyama approximation of one-dimensional stochastic differential equations involving the local time at point zero

  • Mohsine Benabdallah and Kamal Hiderah EMAIL logo

Abstract

We present the Euler–Maruyama approximation for one-dimensional stochastic differential equations involving the local time at point zero. Also, we prove the strong convergence of the Euler–Maruyama approximation whose both drift and diffusion coefficients are Lipschitz. After that, we generalize to the non-Lipschitz case.

References

[1] M. T. Barlow, Skew Brownian motion and a one-dimensional stochastic differential equation, Stochastics 25 (1988), no. 1, 1–2. 10.1080/17442508808833528Search in Google Scholar

[2] R. F. Bass and Z.-Q. Chen, One-dimensional stochastic differential equations with singular and degenerate coefficients, Sankhyā 67 (2005), no. 1, 19–45. Search 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] M. Benabdallah, Y. Elkettani and K. Hiderah, Approximation of Euler-Maruyama for one-dimensional stochastic differential equations involving the local times of the unknown process, Monte Carlo Methods Appl. 22 (2016), no. 4, 307–322. 10.1515/mcma-2016-0115Search in Google Scholar

[5] M. Benabdallah and K. Hiderah, The weak rate of convergence for the Euler–Maruyama approximation of one-dimensional stochastic differential equations involving the local times of the unknown process, preprint (2017), https://arxiv.org/abs/1701.00551v2. Search in Google Scholar

[6] A. Beskos, O. Papaspiliopoulos and G. O. Roberts, Retrospective exact simulation of diffusion sample paths with applications, Bernoulli 12 (2006), no. 6, 1077–1098. 10.3150/bj/1165269151Search in Google Scholar

[7] S. Blei and H.-J. Engelbert, One-dimensional stochastic differential equations with generalized and singular drift, Stochastic Process. Appl. 123 (2013), no. 12, 4337–4372. 10.1016/j.spa.2013.06.014Search in Google Scholar

[8] S. Bouhadou and Y. Ouknine, On the time inhomogeneous skew Brownian motion, Bull. Sci. Math. 137 (2013), no. 7, 835–850. 10.1016/j.bulsci.2013.02.001Search in Google Scholar

[9] 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

[10] 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. 10.1214/aop/1041903215Search in Google Scholar

[11] R. A. Doney and T. Zhang, Perturbed Skorohod equations and perturbed reflected diffusion processes, Ann. Inst. Henri Poincaré Probab. Stat. 41 (2005), no. 1, 107–121. 10.1016/j.anihpb.2004.03.005Search in Google Scholar

[12] H. J. Engelbert and W. Schmidt, On one-dimensional stochastic differential equations with generalized drift, Stochastic Differential Systems (Marseille-Luminy 1984), Lect. Notes Control Inf. Sci. 69, Springer, Berlin (1985), 143–155. 10.1007/BFb0005069Search in Google Scholar

[13] P. Étoré, On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients, Electron. J. Probab. 11 (2006), 249–275. 10.1214/EJP.v11-311Search in Google Scholar

[14] P. Étoré and M. Martinez, On the existence of a time inhomogeneous skew Brownian motion and some related laws, Electron. J. Probab. 17 (2012), 1–27. 10.1214/EJP.v17-1858Search in Google Scholar

[15] P. Étoré and M. Martinez, Exact simulation of one-dimensional stochastic differential equations involving the local time at zero of the unknown process, Monte Carlo Methods Appl. 19 (2013), no. 1, 41–71. 10.1515/mcma-2013-0002Search in Google Scholar

[16] P. Étoré and M. Martinez, Time inhomogeneous stochastic differential equations involving the local time of the unknown process, and associated parabolic operators, Stochastic Process. Appl. 128 (2018), no. 8, 2642–2687. 10.1016/j.spa.2017.09.018Search in Google Scholar

[17] 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

[18] J. M. Harrison and L. A. Shepp, On skew Brownian motion, Ann. Probab. 9 (1981), no. 2, 309–313. 10.1214/aop/1176994472Search in Google Scholar

[19] 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

[20] 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

[21] 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

[22] J.-F. Le Gall, Applications du temps local aux équations différentielles stochastiques unidimensionnelles, Seminar on Probability. XVII, Lecture Notes in Math. 986, Springer, Berlin (1983), 15–31. 10.1007/BFb0068296Search in Google Scholar

[23] J.-F. Le Gall, One-dimensional stochastic differential equations involving the local times of the unknown process, Stochastic Analysis and Applications (Swansea 1983), Lecture Notes in Math. 1095, Springer, Berlin (1984), 51–82. 10.1007/BFb0099122Search in Google Scholar

[24] A. Lejay, On the constructions of the skew Brownian motion, Probab. Surv. 3 (2006), 413–466. 10.1214/154957807000000013Search in Google Scholar

[25] D. Lépingle, Euler scheme for reflected stochastic differential equations, Math. Comput. Simulation 38 (1995), no. 1–3, 119–126. 10.1016/0378-4754(93)E0074-FSearch in Google Scholar

[26] 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

[27] Y. Ouknine, Le “Skew-Brownian motion” et les processus qui en dérivent, Teor. Veroyatnost. i Primenen. 35 (1990), no. 1, 173–179. Search in Google Scholar

[28] Y. Ouknine, F. Russo and G. Trutnau, On countably skewed Brownian motion with accumulation point, Electron. J. Probab. 82 (2013), 1–27. Search in Google Scholar

[29] A. Pilipenko, An Introduction to Stochastic Differential Equations with Reflection, Lectures Pure Appl. Math. 1, Potsdam University, Potsdam, 2014. Search in Google Scholar

[30] N. I. Portenko, Generalized Diffusion Processes (in Russian), “Naukova Dumka”, Kiev, 1982. Search in Google Scholar

[31] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, Berlin, 2005. Search in Google Scholar

[32] A. Semrau, Euler’s approximations of weak solutions of reflecting SDEs with discontinuous coefficients, Bull. Pol. Acad. Sci. Math. 55 (2007), no. 2, 171–182. 10.4064/ba55-2-8Search in Google Scholar

[33] L. Sł omiński, Euler’s approximations of solutions of SDEs with reflecting boundary, Stochastic Process. Appl. 94 (2001), no. 2, 317–337. 10.1016/S0304-4149(01)00087-4Search in Google Scholar

[34] D. W. Stroock and M. Yor, Some remarkable martingales, Seminar on Probability. XV (Strasbourg 1979/1980), Lecture Notes in Math. 850, Springer, Berlin (1981), 590–603. 10.1007/BFb0088396Search in Google Scholar

[35] 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

[36] W. Yue and T. Zhang, Absolute continuity of the laws of perturbed diffusion processes and perturbed reflected diffusion processes, J. Theoret. Probab. 28 (2015), no. 2, 587–618. 10.1007/s10959-013-0499-7Search in Google Scholar

Received: 2018-01-24
Accepted: 2018-09-12
Published Online: 2018-10-09
Published in Print: 2018-12-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 3.3.2025 from https://www.degruyter.com/document/doi/10.1515/mcma-2018-2021/html
Scroll to top button