Convergence and stability of split-step theta methods with variable step-size for stochastic pantograph differential equations

Asymptotical mean-square stability of split-step θ methods for stochastic pantograph differential equations under fully-geometric mesh

“…Assuming that x n is F t n -measurable at the mesh-point t n , we can easily know that x * n and x * n−m are F t n -measurable at related mesh points. Since the convergence of split-step θ methods for SPDEs is investigated in [21], we are interested in the asymptotical mean-square stability of split-step θ methods. Definition 3.2.…”

“…By the inequality (4), we reformulate the conditions in [21], which is a key point in the following analysis.…”

“…Remark 3.6. In [21], the stability conditions for nonlinear system have been discussed, which can be rewritten for linear system as θ > 1 2 and q(2a + b) − |b| − (1 + q)(c 2 + d 2 + cd) > 0 (with our notations). The stability conditions are improved in the Theorem 3.5.…”

“…where ∆ Ñn = Ñn+1 − Ñn , which is called the compensated split-step θ methods. The convergence can be proved with the same method in [21]. Here, we only consider the numerical stability.…”

“…In recently years, a lot work have been done about split-step θ methods. Xiao et al investigated the convergence and the mean-square stability of split-step θ methods in [21]. Guo and Li considered the sufficient conditions of the almost sure stability with general decay rate for the split-step θ methods in [8].…”