Approximation of Euler–Maruyama for one-dimensional stochastic differential equations involving the local times of the unknown process

References

[1] Avikainen R., On irregular functionals of SDEs and the Euler scheme, Finance Stoch. 13 (2009), 381–401. 10.1007/s00780-009-0099-7Search in Google Scholar

[2] Bally V. and Talay D., The law of the Euler scheme for stochastic differential equations. (I): Convergence rate of the distribution function, Probab. Theory Related Fields 104 (1996), 43–60. 10.1007/BF01303802Search in Google Scholar

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

[4] Berkaoui A., Euler scheme for solutions of stochastic differential equations with non-Lipschitz coefficients, Port. Math. (N.S.) 61 (2004), no. 4, 461–478. Search in Google Scholar

[5] Berkaoui A., Bossy M. and Diop A., Euler scheme for SDEs with non-Lipschitz diffusion coefficient: Strong convergence, ESAIM Probab. Stat. 12 (2008), 1–11. 10.1051/ps:2007030Search in Google Scholar

[6] Blei S. and Engelbert H. J., 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

[7] Bouhadou S. and Ouknine Y., 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

[8] Caballero M. E., Fernández B. and Nualart D., Estimation of densities and applications, J. Theoret. Probab. 11 (1998), 831–851. 10.1023/A:1022614917458Search in Google Scholar

[9] Chan K. S. and Stramer O., Weak consistency of the Euler method for numerically solving stochastic differential equations with discontinuous coefficients, Stochastic Process. Appl. 76 (1998), 33–44. 10.1016/S0304-4149(98)00020-9Search in Google Scholar

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

[11] Etoré P. and Martinez M., On the existence of a time inhomogeneous skew Brownian motion and some related laws, Electron. J. Probab. 17 (2012), no. 19, 1–27. 10.1214/EJP.v17-1858Search in Google Scholar

[12] Friedman A., Partial Differential Equations of Parabolic Type, Prentice-Hall, New York, 1964. Search in Google Scholar

[13] Gyöngy I. and Rásonyi M., A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients, Stochastic Process. Appl. 121 (2011), 2189–2200. 10.1016/j.spa.2011.06.008Search in Google Scholar

[14] Harrison J. M. and Shepp L. A., On skew Brownian motion, Ann. Probab. 9 (1981), 309–313. 10.1214/aop/1176994472Search in Google Scholar

[15] Higham D. J., Mao X. and Stuart A. M., Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40 (2002), 1041–1063. 10.1137/S0036142901389530Search in Google Scholar

[16] Kaczor W. J. and Nowak M. T., Problems in Mathematical Analysis: Integration, American Mathematical Society, Providence, 2000. 10.1090/stml/004Search in Google Scholar

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

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

[19] Le Gall J. F., Applications du temps local aux équations différentielles stochastiques unidimensionnelles, Séminaire de Probabilités XVII 1981/82, Lecture Notes in Math. 986, Springer, Berlin (1983), 15–31. 10.1007/BFb0068296Search in Google Scholar

[20] Le Gall J. F., 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

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

[22] Lesnic D., Characterizations of the functions with bounded variation, Acta Univ. Apulensis Math. Inform. 6 (2003), 47–54. Search in Google Scholar

[23] Ngo H.-L. and Taguchi D., 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

[24] Ouknine Y., “Skew-Brownian motion” and derived processes, Theory Probab. Appl. 35 (1990), 163–169. 10.1137/1135018Search in Google Scholar

[25] Ouknine Y., Trutnau G. and Russo F., On countably skewed Brownian motion with accumulation point, Electron. J. Probab. 20 (2015), Paper No. 82. 10.1214/EJP.v20-3640Search in Google Scholar

[26] Portenko N. I., Generalized Diffusion Processes (in Russian), Naukova Dumka, Kiev, 1982; translation published by American Mathematical Society, Providence, 1990. 10.1090/mmono/083Search in Google Scholar

[27] Stroock D. W. and Yor M., Some remarkable martingales, Séminaire de Probabilités XV 1979/80, Lecture Notes in Math. 850, Springer, Berlin (1981), 590–603. 10.1007/BFb0088396Search in Google Scholar

[28] Taguchi D., Lp⁢(Ω)-difference of one-dimensional stochastic differential equations with discontinuous drift, preprint 2014, https://arxiv.org/abs/1404.2358v1. Search in Google Scholar

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

[30] Yan B. L., The Euler scheme with irregular coefficients, Ann. Probab. 30 (2002), 1172–1194. 10.1214/aop/1029867124Search in Google Scholar