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

Zakaria Ali, Minyahil Abera Abebe and Talat Nazir ()
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Zakaria Ali: Department of Mathematical Sciences, University of South Africa, Florida 0003, South Africa
Minyahil Abera Abebe: Department of Mathematical Sciences, University of South Africa, Florida 0003, South Africa
Talat Nazir: Department of Mathematical Sciences, University of South Africa, Florida 0003, South Africa

Mathematics, 2024, vol. 12, issue 18, 1-36

Abstract: 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.

Keywords: variable-order Caputo–Fabrizio fractional stochastic differential equation; non-singular non-local effect; Euler–Maruyama method; strong convergence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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