On mean-square stability properties of a new adaptive stochastic Runge–Kutta method

We analyze the mean-square (MS) stability properties of a newly introduced adaptive time-stepping stochastic Runge–Kutta method which relies on two local error estimators based on drift and diffusion terms of the equation [A. Foroush Bastani, S.M. Hosseini, A new adaptive Runge–Kutta method for stochastic differential equations, J. Comput. Appl. Math. 206 (2007) 631–644]. In the same spirit as [H. Lamba, T. Seaman, Mean-square stability properties of an adaptive time-stepping SDE solver, J. Comput. Appl. Math. 194 (2006) 245–254] and with applying our adaptive scheme to a standard linear multiplicative noise test problem, we show that the MS stability region of the adaptive method strictly contains that of the underlying stochastic differential equation. Some numerical experiments confirms the validity of the theoretical results.