Euler-Maruyama Scheme for SDEs with Dini Continuous Coefficients
Issue:
Volume 9, Issue 2, June 2023
Pages:
18-25
Received:
29 May 2023
Accepted:
26 June 2023
Published:
6 July 2023
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.
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...
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Facrorization of Symmetric and Obliquely Symmetric Polynomials
Rena Eldar Kizi Kerbalayeva
Issue:
Volume 9, Issue 2, June 2023
Pages:
26-29
Received:
5 June 2022
Accepted:
25 June 2023
Published:
8 July 2023
Abstract: Meeting some mathematical and algebraic challenges is going to need more mathematical branches to get involved, new ways for mathematical branches interact with other methods and new rules of funding for mathematical paper importantly. The area is so broad that, this paper is not able possibly obtain every problem, but it gives a number of representative examples that the along of this paper that factorizations to be done. I must note that, Algebra is not only a major subject of science, but is also interesting and difficult. This paper is important, not just for Algebra, but for all fields related to mathematics. In addition, factorization of polynomial is one of important and using concept of mathematics. In present paper symmetric and obliquely symmetric polynomials, based on factorization concept have been studied. Furthermore, several integral steps associated with the considered polynomials both of symmetric and obliquely symmetric polynomials type has been recently introduced and in addition factorization of such polynomials have been studied. In this paper I introduce two new and different uses of factorization of symmetric and symmetric polynomials: first we study symmetric polynomials, then we study obliquely symmetric polynomials and we also look through the new idea for factorizations of such type polynomials.
Abstract: Meeting some mathematical and algebraic challenges is going to need more mathematical branches to get involved, new ways for mathematical branches interact with other methods and new rules of funding for mathematical paper importantly. The area is so broad that, this paper is not able possibly obtain every problem, but it gives a number of represen...
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