Convergence and stability of the one-leg θ method for stochastic differential equations with piecewise continuous arguments

The equivalent relation is established here about the stability of stochastic differential equations with piecewise continuous arguments(SDEPCAs) and that of the one-leg θ method applied to the SDEPCAs. Firstly, the convergence of the one-leg θ method to SDEPCAs under the global Lipschitz condition is proved. Secondly, it is proved that the SDEPCAs are pth(p ∈ (0, 1)) moment exponentially stable if and only if the one-leg θ method is pth moment exponentially stable for some sufficiently small step-size. Thirdly, the corollaries that the pth moment exponential stability of the SDEPCAs (the one-leg θ method) implies the almost sure exponential stability of the SDEPCAs (the one-leg θ method) are given. Finally, numerical simulations are provided to illustrate the theoretical results.