Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations

In this paper the sufficient conditions of existence and uniqueness of the solutions for stochastic pantograph equation are given, i.e., the local Lipschitz condition and the linear growth condition. Under the Lipschitz condition and the linear growth condition it is proved that the semi-implicit Euler method is convergence with strong order 1/2. © 2006 Elsevier Inc. All rights reserved.