The convergence of the split-step backward Euler (SSBE) method applied to stochastic differential equation with variable delay is proven inLp-sense. Almost sure convergence is derived from theLpconvergence by Chebyshev’s inequality and the Borel-Cantelli lemma; meanwhile, the convergence rate is obtained.
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.
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