Convergence and stability of one-leg $\theta$ method for stochastic differential equations with piecewise continuous arguments

Abstract

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