Asymptotical mean-square stability of linear θ-methods for stochastic pantograph differential equations: variable stepsize and transformation approach

The paper deals with the asymptotical mean-square stability of the linear θ-methods under variable stepsize and transformation approach for stochastic pantograph differential equations. A limiting equation for the analysis of numerical stability is introduced by Kronecker products. Under the condition which guarantee the stability of exact solutions, the optimal stability region of the linear θ-methods under variable stepsize is given by using the limiting equation, i.e. (Formula presented.), which is the same to the deterministic problems. Moreover, the linear θ-methods under the transformation approach are also considered and the result of the stability is improved for (Formula presented.). Finally, numerical examples are given to illustrate the asymptotical mean-square stability under variable stepsize and transformation approach.