We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations, which generalize the free Euler-Maruyama method introduced by Schl\"{u}chtermann and Wibmer [27]. Under some mild conditions, we prove the strong convergence and exponential stability in mean square of the numerical solution. The free stochastic theta method with $\theta=1$ can inherit the exponential stability of original equations for any given step size. Our method can offer better stability and efficiency than the free Euler-Maruyama method. Moreover, numerical results are reported to confirm these theoretical findings.

翻译：我们引入了自由随机微分方程随机theta方法的自由概率类比，这类方法推广了Schlüchtermann和Wibmer [27]提出的自由Euler-Maruyama方法。在温和条件下，我们证明了数值解的强收敛性和均方指数稳定性。对于任意给定步长，参数θ=1的自由随机theta方法能够继承原方程的指数稳定性。我们的方法相比自由Euler-Maruyama方法具有更好的稳定性和效率。此外，数值实验结果验证了这些理论发现。