Almost Sure and Lp Convergence of Split-Step Backward Euler Method for Stochastic Delay Differential Equation

Abstract

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.