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Asymptotical mean-square stability of linear θ-methods for stochastic pantograph differential equations: variable stepsize and transformation approach
  • xiaochen yang,
  • Zhanwen Yang,
  • Yu Xiao
xiaochen yang
Harbin Institute of Technology
Author Profile
Zhanwen Yang
Harbin Institute of Technology
Author Profile
Yu Xiao
Harbin Institute of Technology
Author Profile

Abstract

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., θ ∈ (1/ 2,1], 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 θ = 1 /2. Finally, numerical examples are given to illustrate the asymptotical meansquare stability under variable stepsize and transformation approach.