Construction of New Implicit Milstein-Type Scheme for Solution of the Systems of SODEs

The main aim of this study is to construct a new approximation method for solving stif Itˆo stochastic ordinary diferential equations based on explicit Milstein scheme. Under the suicient conditions such as Lipschitz condition and linear growth bound, we prove that split-step (α, β)-Milstein method strongly convergence to exact solution with order one. The means-quare stability of the our method for linear stochastic diferential equation with one-dimensional Wiener process is studied. Stability analysis shows that the mean-square stability our proposed method contains the mean-square stability region of linear scalar test equation. Finally, numerical examples illustrates the efectiveness of the theoretical results.