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Almost sure exponential stability of the split-step backward Euler (SSBE) method applied to an Itô-type stochastic differential equation with time-varying delay is discussed by the techniques based on Doob-Mayer decomposition and semimartingale convergence theorem. Numerical experiments confirm the theoretical analysis.

The Semimartingale Approach to Almost Sure Stability Analysis of a Two-Stage Numerical Method for Stochastic Delay Differential Equation