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This paper aims to present a new pathwise approximation method, which gives approximate solutions of order3  2$\begin{array}{} \displaystyle \frac{3}{2} \end{array}$    for stochastic differential equations (SDEs) driven by multidimensional Brownian motions. The new method, which assumes the diffusion matrix non-degeneracy, employs the Runge-Kutta method and uses the Itô-Taylor expansion, but the generating of the approximation of the expansion is carried out as a whole rather than individual terms. The new idea we applied in this paper is to replace the iterated stochastic integrals  I α   by random variables, so implementing this scheme does not require the computation of the iterated stochastic integrals  I α  . Then, using a coupling which can be found by a technique from optimal transport theory would give a good approximation in a mean square. The results of implementing this new scheme by MATLAB confirms the validity of the method.

Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion