M. H. Song

M. H. Song, Y. L. Lu, M. Z. Liu

Recently, Martin Hutzenthaler pointed out that the explicit Euler method fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with superlinearly growing and globally one sided Lipschitz drift coefficient. Afterwards, he proposed an explicit and easily implementable Euler method, i.e tamed Euler method, for such an SDE and showed that this method converges strongly with order of one half. In this paper, we use the tamed Euler method to solve the stochastic differential equations with piecewise continuous arguments (SEPCAs) with superlinearly growing coefficients and prove that this method is convergent with strong order one half.

M. H. Song, Y. L. Lu, M. Z. Liu

In this paper, we study the polynomial stability of analytical solution and convergence of the semi-implicit Euler method for non-linear stochastic pantograph differential equations. Firstly, the sufficient conditions for solutions to grow at a polynomial rate in the sense of mean-square and almost surely are obtained. Secondly, the consistence and convergence of this method are proved. Furthermore, the orders of consistence (in the sense of average and mean-square) and convergence are given, respectively.