Mean-square stability of a constructed Third-order stochastic Runge--Kutta schemes for general stochastic differential equations

In this paper, we are interested in construction of an explicit third-order stochastic Runge–Kutta (SRK3) schemes for the weak approximation of stochastic differential equations (SDEs) with the general diffusion coefficient b(t, x). To this aim, we use the Itˆo-Taylor method and compare them with the stochastic expansion of the approximation. In this way, the authors encountered with a large number of equations and could find to derive four families for SRK3 schemes. Also we investigate the mean-square stability (MS-stability) properties of SRK3 schemes for a linear SDE. Finally, the proposed families are implemented on some examples to illustrate convergence results.