# Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler--Maruyama approximation

## TL;DR

This paper studies nonlinear stochastic fractional integro-differential equations with singular kernels, establishing well-posedness and analyzing the strong convergence and superconvergence of the Euler--Maruyama numerical method.

## Contribution

It provides the first detailed analysis of existence, uniqueness, and numerical approximation for equations with Abel-type singular kernels.

## Key findings

- Proved existence, uniqueness, and continuous dependence of solutions.
- Developed and analyzed the Euler--Maruyama method for these equations.
- Achieved strong first-order superconvergence when =1.

## Abstract

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.