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Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients

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.

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

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

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

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

Copyright © 2023 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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