Strong Convergence of Euler-Type Methods for Nonlinear Fractional Stochastic Differential Equations without Singular Kernel

In this paper, we first prove the existence and uniqueness of the solution to a variable-order Caputo–Fabrizio fractional stochastic differential equation driven by a multiplicative white noise, which describes random phenomena with non-local effects and non-singular kernels. The Euler–Maruyama sche...

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Bibliographic Details
Published in:	Mathematics (Basel) Vol. 12; no. 18; p. 2890
Main Authors:	Ali, Zakaria, Abebe, Minyahil Abera, Nazir, Talat
Format:	Journal Article
Language:	English
Published:	Basel MDPI AG 01-09-2024
Subjects:	Analysis Applied mathematics Approximation Convergence Convergence (Mathematics) Differential equations Differential equations, Nonlinear Euler–Maruyama method Integral calculus Investigations Kernel functions Lipschitz condition Noise intensity non-singular non-local effect Partial differential equations Physics Singularities (Mathematics) Stochastic models Stochastic processes strong convergence variable-order Caputo–Fabrizio fractional stochastic differential equation White noise South Africa
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Summary:	In this paper, we first prove the existence and uniqueness of the solution to a variable-order Caputo–Fabrizio fractional stochastic differential equation driven by a multiplicative white noise, which describes random phenomena with non-local effects and non-singular kernels. The Euler–Maruyama scheme is extended to develop the Euler–Maruyama method, and the strong convergence of the proposed method is demonstrated. The main difference between our work and the existing literature is the fact that our assumptions on the nonlinear external forces are those of one-sided Lipschitz conditions on both the drift and the nonlinear intensity of the noise as well as the proofs of the higher integrability of the solution and the approximating sequence. Finally, to validate the numerical approach, current results from the numerical implementation are presented to test the efficiency of the scheme used in order to substantiate the theoretical analysis.
ISSN:	2227-7390 2227-7390
DOI:	10.3390/math12182890