I'm using a 4th order Adams predictor-corrector method to numerically solve a regular differential equation. Now I would be interested to be able to include a noisy term to the equation -as in the Euler-Maruyama method, the classical and easy way to simulate a Brownian motion via a Wiener process-, but I have almost zero experience with convergence and orders in Monte Carlo methods.
Is there a canonical way to do it? I believe the Runge-Kutta method can be adapted to stochastic differential equations but, as before, I know next to nothing about it. As a first guess, I would include a $w_{0,1}\cdot \sigma \cdot \sqrt{h}$ term in the corrector part.
Could you give me any reference or describe an example on how to do it?