Stochastic theta methods for free stochastic differential equations

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.