Almost sure and \(L^p\) convergence of split-step backward Euler method for stochastic delay differential equation (Q1724003)

Summary: The convergence of the split-step backward Euler (SSBE) method applied to stochastic differential equation with variable delay is proven in \(L^p\)-sense. Almost sure convergence is derived from the \(L^p\) convergence by Chebyshev's inequality and the Borel-Cantelli lemma; meanwhile, the convergence rate is obtained.