Numerical solutions of neutral stochastic functional differential equations

This paper examines the numerical solutions of neutral stochastic functional differential equations (NSFDEs) d[x(t) - u(xt)] = f(xt)dt + g(xt)dw(t), t >= 0. The key contribution is to establish the strong mean square convergence theory of the Euler-Maruyama approximate solution under the local Lipschitz condition, the linear growth condition, and contractive mapping. These conditions are generally imposed to guarantee the existence and uniqueness of the true solution, so the numerical results given here are obtained under quite general conditions. Although the way of analysis borrows from [ X. Mao, LMS J. Comput. Math., 6 (2003), pp. 141 - 161], to cope with u( xt), several new techniques have been developed.