Stability analysis of two classes of θ-Milstein schemes for stochastic differential delay equations

In this paper we introduce a split-step θ-Milstein (SSTM) method for stochastic differential delay equations (SDDEs). For θ ∈ [0; 1/2 ], these two classes of θ-Milstein schemes can share the exponential mean-square stability of the exact solution. But for θ ∈ ( 1/2 ; 1], under the additional linear growth condition, this method can also reproduce the exponential mean-square stability of the exact solution. Then, based on the stability results of SSTM scheme, we examine the exponential mean-square stability of the SSTM approximation for SDDEs.