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 $θ\in [1/2, 1]$ 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.
Convergence and stability of two classes of theta-Milstein schemes for stochastic differential equations
Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院系统科学研究所）；School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；School of Computer Science and Technology, Huazhong University of Science and Technology（华中科技大学计算机科学与技术学院）
Weak order one convergence of structure-preserving stochastic theta methods for stochastic differential algebraic equations with time-dependent singular matrices
Yunnan University（云南大学）；Jiangxi University of Finance and Economics（江西财经大学）
Strong convergence rates of stochastic theta Milstein methods for index-1 stochastic differential algebraic equations under non-globally Lipschitz conditions
School of Mathematics and Statistics, Yunnan University（云南大学数学与统计学院）；School of Statistics and Data Science, Jiangxi University of Finance and Economics（江西财经大学统计与数据科学学院）
Strong convergence and Mittag-Leffler stability of stochastic theta method for time-changed stochastic differential equations
Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心，清华大学）；Department of Mathematical Sciences, Tsinghua University（数学系，清华大学）；School of Economics and Management, Tsinghua University（经济管理学院，清华大学）
Stability equivalence among stochastic differential equations and stochastic differential equations with piecewise continuous arguments and corresponding Euler-Maruyama methods
