We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations in this work. Under some mild conditions, we prove the strong convergence and exponential stability in mean square of the numerical methods. The free stochastic theta method with $\theta=1$ can inherit the exponential stability of original equations for any given step size. Our method offers better stability and efficiency than the free Euler-Maruyama method. Moreover, numerical results are reported to confirm these theoretical findings.

翻译：本文引入了自由随机微分方程的随机θ方法的自由概率类比。在温和条件下，我们证明了数值方法的均方强收敛性与指数稳定性。当θ=1时，自由随机θ方法对任意给定步长均可继承原方程的指数稳定性。本方法较自由Euler-Maruyama方法具有更好的稳定性与计算效率。数值实验结果进一步验证了理论结论。