This manuscript examines the problem of nonlinear stochastic fractional neutral integro-differential equations with weakly singular kernels. Our focus is on obtaining precise estimates to cover all possible cases of Abel-type singular kernels. Initially, we establish the existence, uniqueness, and continuous dependence on the initial value of the true solution, assuming a local Lipschitz condition and linear growth condition. Additionally, we develop the Euler-Maruyama method for the numerical solution of the equation and prove its strong convergence under the same conditions as the well-posedness. Moreover, we determine the accurate convergence rate of this method under global Lipschitz conditions and linear growth conditions. And also we have proven generalized Gronwall inequality with a multi-weakly singularity.

翻译：本文研究具有弱奇异核的非线性随机分数阶中立型积分微分方程问题。我们的重点在于获取能够涵盖阿贝尔型奇异核所有可能情况的精确估计。首先，在假设局部利普希茨条件和线性增长条件的情况下，我们建立了真解的存在性、唯一性及对初值的连续依赖性。此外，我们针对该方程发展了欧拉-丸山数值解法，并在与适定性相同的条件下证明了其强收敛性。同时，在全局利普希茨条件和线性增长条件下，我们确定了该方法的精确收敛速率。此外，我们还证明了具有多重弱奇异性的广义格朗沃尔不等式。