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