Convergence and stability of the split-step θθ-method for stochastic differential equations

In this paper, we construct a new split-step method for solving stochastic differential equations, namely the split-step θθ-method. Under Lipschitz and linear growth conditions, we establish a mean-square convergence theory of split-step θθ-approximate solutions. Moreover, the mean-square stability of the method for a linear test equation with real parameters is considered and the real mean-square stability region is plotted. Finally, numerical results are presented to demonstrate the efficiency of the split-step θθ-method.