Geodesic
Convergence analysis of semi-implicit Euler method for nonlinear stochastic delay differential equations of neutral type
Serdica Mathematical Journal, Tome 43 (2017) no. 2, pp. 147-160.

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The main purpose of this paper is to study the convergence of numerical solutions to a class of neutral stochastic delay differential equations (NSDDEs) in Itô sense. The basic idea is to reformulate the original problem eliminating the dependence on the differentiation of the solution in the past values, which leads to a stochastic differential algebraic system. It is shown that the Semi-implicit Euler (SIE) method with two parameters θ and λ is mean-square convergent with order p =1/2 for Lipschitz continuous coefficients of underlying NSDDEs. A nonlinear numerical example illustrates the theoretical results.
Keywords: Neutral stochastic delay, differential equations, mean-square convergence, semiimplicit Euler method, 65C20, 60H35, 65C30 
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     title = {Convergence analysis of semi-implicit {Euler} method for nonlinear stochastic delay differential equations of neutral type},
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Milev, Marian; Rouz, O. Farkhondeh; Ahmadian, D. Convergence analysis of semi-implicit Euler method for nonlinear stochastic delay differential equations of neutral type. Serdica Mathematical Journal, Tome 43 (2017) no. 2, pp. 147-160. http://geodesic.mathdoc.fr/item/SMJ2_2017_43_2_a5/