# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021225
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## Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation

 School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan 411105, China

* Corresponding author: Aiguo Xiao

Received  May 2020 Revised  April 2021 Early access September 2021

Fund Project: This research is supported by the National Natural Science Foundation of China (Nos. 12071403, 11671343), the Research Foundation of Education Commission of Hunan Province of China (No. 19B565) and the Postgraduate Innovation Fund of Hunan Province in China (No. CX20190420)

This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler–Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler–Maruyama method can reach strong first-order superconvergence when $\alpha = 1$. Finally, several numerical tests are reported for verification of the theoretical findings.

Citation: Xinjie Dai, Aiguo Xiao, Weiping Bu. Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021225
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##### References:
The mean square errors of the EM scheme (11) for Example 5.1
The mean square errors of the EM scheme (11) for Example 5.2
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