A note on convergence of semi-implicit Euler methods for stochastic pantograph equations

In the literature [1] [Existence and uniqueness of the solutions and convergence of semiimplicit Euler methods for stochastic pantograph equation, J. Math. Anal. Appl. 325 (2007) 1142 1159], Fan and Liu investigated the existence and uniqueness of the solution for stochastic pantograph equation and proved the convergence of the semi-implicit Euler methods under the Lipschitz condition and the linear growth condition. Unfortunately, the main result of convergence derived by the conditions is somewhat restrictive for the purpose of practical application, because there are many stochastic pantograph equations that only satisfy the local Lipschitz condition. In this note we improve the corresponding results in the above-mentioned reference.