Delay dependent stability of stochastic split-step θ methods for stochastic delay differential equations

In this paper, the delay dependent asymptotic mean square stability of the stochastic split-step θ method for a scalar linear stochastic delay differential equation with real coefficients is studied. The full stability region of this method is given by using root locus technique. The necessary and sufficient condition with respect to the equation coefficients, time stepsize and method parameter θ is derived. It is proved that the stochastic split-step backward Euler can preserve the asymptotic mean square stability of the underlying system completely. In addition, the numerical stability regions of the stochastic split-step θ method and the stochastic θ method are compared with each other. At last, we validate our conclusions by numerical experiments.