Strong and weak divergence of the backward Euler method for neutral stochastic differential equations with time-dependent delay

Abstract

. In this article, we consider the backward Euler method for a class of neutral stochastic differential equations with time-dependent delay and reveal conditions of the divergence of the p-th absolute moments of the corresponding approximate solutions when p ∈ ( 0 , ∞ ) . This implies the strong L p -divergence of the method at finite time, for p ∈ [ 1 , + ∞ ) , and shows that its numerically weak convergence fails to hold. This article is motivated by the paper Hutzenthaler, M., Jentzen, A., Kloeden, P. E.: Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. A 467 (2011), no. 2130, 1563–1576, where a class of ordinary stochastic differential equations with superlinearly growing coefficients is studied.