Multi-colored rooted tree analysis of the weak order conditions of a stochastic Runge–Kutta family

The aim of the present paper is to give a tractable way of seeking weak order conditions of a stochastic Runge–Kutta family for stochastic differential equations with a multi-dimensional Wiener process. This is accomplished by an extension of the rooted tree analysis for ordinary Runge–Kutta methods. As a result, weak order conditions can be obtained directly from diagrams of multi-colored rooted trees. This new methodology will be of benefit when high weak order conditions need to be sought.