Stochastic theta methods for free stochastic differential equations

Abstract: We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations in this work. Assume that the drift coefficient of the free stochastic differential equations is operator Lipschitz and the diffusion coefficients are locally operator Lipschitz, we prove the strong convergence of the numerical methods. Moreover, we investigate the exponential stability in mean square of the equations and the numerical methods. In particular, the free stochastic theta methods with $\theta \in [1/2, 1]$ can inherit the exponential stability of original equations for any given step size. Our methods offer better stability than the free Euler-Maruyama method. Numerical results are reported to confirm these theoretical findings and show the efficiency of our methods compared with the free Euler-Maruyama method.