The novelty of this paper is to derive a mild solution by means of recently defined Mittag-Leffler type functions of fractional stochastic Langevin equations of orders $\alpha \in (1,2]$ and $\beta \in (0,1]$ whose coefficients satisfy standard Lipschitz and linear growth conditions. Then, we prove existence and uniqueness results of mild solution and show the coincidence between the notion of mild solution and integral equation. For this class of system, we construct fractional Euler–Maruyama method and establish new results on strong convergence of this method for fractional stochastic Langevin equations. We also introduce a general form of the nonlinear fractional stochastic Langevin equation and derive a general mild solution. Finally, the numerical examples are illustrated to verify the main theory.

本文的新颖之处在于通过最近定义的分数阶随机朗之万方程的 Mittag-Leffler 型函数推导出温和解 $\alpha \in (1,2]$ 和 $\beta \in (0,1]$其系数满足标准 Lipschitz 和线性增长条件。然后，证明了温和解的存在唯一性结果，证明了温和解的概念与积分方程的一致性。对于这类系统，我们构造了分数阶 Euler-Maruyama 方法，并建立了该方法对分数阶随机朗之万方程的强收敛性的新结果。我们还介绍了非线性分数阶随机朗之万方程的一般形式并推导出一般温和解。最后通过算例对主要理论进行验证。