Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients

Zhen Wang

doi:doi:10.11648/j.ml.20230902.11

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Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients

Zhen Wang

Published in Mathematics Letters (Volume 9, Issue 2)

Received: 29 May 2023 Accepted: 26 June 2023 Published: 6 July 2023

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Abstract

In the study of Euler-Maruyama scheme for Stochastic Differential Equations, researchers focus on the convergence rate under different conditions, using analytical methods and Stochastic Partial Differential Equation. One of them is to study the Lipschitz continuous, mainly from drift coefficient and diffusion coefficient. The other is the study of non-Lipschitz continuous, since most of the real life is not Lipschitz continuous. Therefore, most researchers are looking at non-Lipschitz continuous. In my study, without loss of generality, we are also a continuous study of non-Lipschitz and a faster convergence rate. In this paper, we show the convergence rate of Euler-Maruyama scheme for non-degenerate SDEs where the drift term b and the diffusion term σ are the uniformly bounded, b and σ satisfy correlated conditions of Dini-continuous, by the aid of the regularity of the solution to the associated Kolmogorov equation of SPDE and common methods in stochastic analysis, including Itô’s formula, Jensen’s inequality, Hölder inequality BDG’s inequality, Gronwall’s inequality. We obtain the same conclusions by weakening the conditions of previous research using the properties of Dini continuous and Taylor expansion. At the same time, we also reached the same conclusion under local boundedness and local Dini-continuous. Moreover, my research results have laid the groundwork for the follow-up research.

Published in	Mathematics Letters (Volume 9, Issue 2)
DOI	10.11648/j.ml.20230902.11
Page(s)	18-25
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

Non-Degenerate, Stochastic Differential Equation, Euler-Maruyama Scheme, Dini Continuous, Kolmogorov Equation

References

[1]	J. Bao, X. Huang and C. Yuan, Convergence rate of Euler-Maruyama scheme for SDEs with Hölder-Dini continuous drifts. J. Theoret. Probab.: 32 (2019) 848-871.
[2]	N. Halidias and P. E. Kloeden, A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient. BIT: 48 (2008) 51-59.
[3]	Gyöngy, A note on Euler approximations. Potential Anal.: 8 (1998) 205-216.
[4]	I. Gyöngy and M. Rásonyi, A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients. Stochastic Process. Appl.: 121 (2011) 2189-2200.
[5]	P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1995.
[6]	Leobacher G., Szölgyenyi M. A numerical method for SDEs with discontinuous drift BIT, 56 (1) (2015), 151-162.
[7]	Leobacher G., Szölgyenyi M. A strong order 1∕2 method for multidimensional SDEs with discontinuous drift. Ann. Appl. Probab., 27 (4) (2017), 2383-2418.
[8]	Leobacher G., Szölgyenyi M. Convergence of the Euler –Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient Numer. Math., 138 (1) (2018), 219-239.
[9]	O. Menoukeu Pamen and D. Taguchi, Strong rate of convergence for the Euler-Maruyama approximation of SDEs with Hölder continuous drift coefficient. Stochastic Process. Appl.: 127 (2017) 2542-2559.
[10]	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) 1793-1819.
[11]	H.-L. Ngo and D. Taguchi, On the Euler-Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients. IMA J. Numer. Anal. 37 (2017), no. 4, 1864–1883.
[12]	H.-L. Ngo and D. Taguchi, On the Euler-Maruyama scheme for SDEs with bounded variation and Hölder continuous coefficients. Math. Comput. Simulation: 161 (2019) 102-112.
[13]	L. Yan, The Euler scheme with irregular coefficients. Ann. Probab.: 30 (2002) 1172-1194.
[14]	T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ.: 11 (1971) 155-167.
[15]	X. Zhang, Euler-Maruyama approximations for SDEs with non-Lipschitz coefficients and applications. J. Math. Anal. Appl.: 316 (2006) 447-458.

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APA Style

Zhen Wang. (2023). Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients. Mathematics Letters, 9(2), 18-25. https://doi.org/10.11648/j.ml.20230902.11

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ACS Style

Zhen Wang. Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients. Math. Lett. 2023, 9(2), 18-25. doi: 10.11648/j.ml.20230902.11

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AMA Style

Zhen Wang. Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients. Math Lett. 2023;9(2):18-25. doi: 10.11648/j.ml.20230902.11

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@article{10.11648/j.ml.20230902.11,
  author = {Zhen Wang},
  title = {Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients},
  journal = {Mathematics Letters},
  volume = {9},
  number = {2},
  pages = {18-25},
  doi = {10.11648/j.ml.20230902.11},
  url = {https://doi.org/10.11648/j.ml.20230902.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20230902.11},
  abstract = {In the study of Euler-Maruyama scheme for Stochastic Differential Equations, researchers focus on the convergence rate under different conditions, using analytical methods and Stochastic Partial Differential Equation. One of them is to study the Lipschitz continuous, mainly from drift coefficient and diffusion coefficient. The other is the study of non-Lipschitz continuous, since most of the real life is not Lipschitz continuous. Therefore, most researchers are looking at non-Lipschitz continuous. In my study, without loss of generality, we are also a continuous study of non-Lipschitz and a faster convergence rate. In this paper, we show the convergence rate of Euler-Maruyama scheme for non-degenerate SDEs where the drift term b and the diffusion term σ are the uniformly bounded, b and σ satisfy correlated conditions of Dini-continuous, by the aid of the regularity of the solution to the associated Kolmogorov equation of SPDE and common methods in stochastic analysis, including Itô’s formula, Jensen’s inequality, Hölder inequality BDG’s inequality, Gronwall’s inequality. We obtain the same conclusions by weakening the conditions of previous research using the properties of Dini continuous and Taylor expansion. At the same time, we also reached the same conclusion under local boundedness and local Dini-continuous. Moreover, my research results have laid the groundwork for the follow-up research.},
 year = {2023}
}

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TY  - JOUR
T1  - Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients
AU  - Zhen Wang
Y1  - 2023/07/06
PY  - 2023
N1  - https://doi.org/10.11648/j.ml.20230902.11
DO  - 10.11648/j.ml.20230902.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 18
EP  - 25
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20230902.11
AB  - In the study of Euler-Maruyama scheme for Stochastic Differential Equations, researchers focus on the convergence rate under different conditions, using analytical methods and Stochastic Partial Differential Equation. One of them is to study the Lipschitz continuous, mainly from drift coefficient and diffusion coefficient. The other is the study of non-Lipschitz continuous, since most of the real life is not Lipschitz continuous. Therefore, most researchers are looking at non-Lipschitz continuous. In my study, without loss of generality, we are also a continuous study of non-Lipschitz and a faster convergence rate. In this paper, we show the convergence rate of Euler-Maruyama scheme for non-degenerate SDEs where the drift term b and the diffusion term σ are the uniformly bounded, b and σ satisfy correlated conditions of Dini-continuous, by the aid of the regularity of the solution to the associated Kolmogorov equation of SPDE and common methods in stochastic analysis, including Itô’s formula, Jensen’s inequality, Hölder inequality BDG’s inequality, Gronwall’s inequality. We obtain the same conclusions by weakening the conditions of previous research using the properties of Dini continuous and Taylor expansion. At the same time, we also reached the same conclusion under local boundedness and local Dini-continuous. Moreover, my research results have laid the groundwork for the follow-up research.
VL  - 9
IS  - 2
ER  -

Copy | Download

Author Information

Zhen Wang

College of Mathematics and Information Statistics, Henan Normal University, Xinxiang, China

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Plain Text BibTeX RIS

APA Style

Zhen Wang. (2023). Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients. Mathematics Letters, 9(2), 18-25. https://doi.org/10.11648/j.ml.20230902.11

Copy | Download

ACS Style

Zhen Wang. Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients. Math. Lett. 2023, 9(2), 18-25. doi: 10.11648/j.ml.20230902.11

Copy | Download

AMA Style

Zhen Wang. Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients. Math Lett. 2023;9(2):18-25. doi: 10.11648/j.ml.20230902.11

Copy | Download

@article{10.11648/j.ml.20230902.11,
  author = {Zhen Wang},
  title = {Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients},
  journal = {Mathematics Letters},
  volume = {9},
  number = {2},
  pages = {18-25},
  doi = {10.11648/j.ml.20230902.11},
  url = {https://doi.org/10.11648/j.ml.20230902.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20230902.11},
  abstract = {In the study of Euler-Maruyama scheme for Stochastic Differential Equations, researchers focus on the convergence rate under different conditions, using analytical methods and Stochastic Partial Differential Equation. One of them is to study the Lipschitz continuous, mainly from drift coefficient and diffusion coefficient. The other is the study of non-Lipschitz continuous, since most of the real life is not Lipschitz continuous. Therefore, most researchers are looking at non-Lipschitz continuous. In my study, without loss of generality, we are also a continuous study of non-Lipschitz and a faster convergence rate. In this paper, we show the convergence rate of Euler-Maruyama scheme for non-degenerate SDEs where the drift term b and the diffusion term σ are the uniformly bounded, b and σ satisfy correlated conditions of Dini-continuous, by the aid of the regularity of the solution to the associated Kolmogorov equation of SPDE and common methods in stochastic analysis, including Itô’s formula, Jensen’s inequality, Hölder inequality BDG’s inequality, Gronwall’s inequality. We obtain the same conclusions by weakening the conditions of previous research using the properties of Dini continuous and Taylor expansion. At the same time, we also reached the same conclusion under local boundedness and local Dini-continuous. Moreover, my research results have laid the groundwork for the follow-up research.},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients
AU  - Zhen Wang
Y1  - 2023/07/06
PY  - 2023
N1  - https://doi.org/10.11648/j.ml.20230902.11
DO  - 10.11648/j.ml.20230902.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 18
EP  - 25
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20230902.11
AB  - In the study of Euler-Maruyama scheme for Stochastic Differential Equations, researchers focus on the convergence rate under different conditions, using analytical methods and Stochastic Partial Differential Equation. One of them is to study the Lipschitz continuous, mainly from drift coefficient and diffusion coefficient. The other is the study of non-Lipschitz continuous, since most of the real life is not Lipschitz continuous. Therefore, most researchers are looking at non-Lipschitz continuous. In my study, without loss of generality, we are also a continuous study of non-Lipschitz and a faster convergence rate. In this paper, we show the convergence rate of Euler-Maruyama scheme for non-degenerate SDEs where the drift term b and the diffusion term σ are the uniformly bounded, b and σ satisfy correlated conditions of Dini-continuous, by the aid of the regularity of the solution to the associated Kolmogorov equation of SPDE and common methods in stochastic analysis, including Itô’s formula, Jensen’s inequality, Hölder inequality BDG’s inequality, Gronwall’s inequality. We obtain the same conclusions by weakening the conditions of previous research using the properties of Dini continuous and Taylor expansion. At the same time, we also reached the same conclusion under local boundedness and local Dini-continuous. Moreover, my research results have laid the groundwork for the follow-up research.
VL  - 9
IS  - 2
ER  -

Copy | Download