# Numerical method for SDEs

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?

• A standard reference in numerical methods for SDE:s is the book by Kloeden and Platen, Numerical Solution of Stochastic Differential Equations. (Yes, "Solution" is written in singular in the title) – A.Sh Dec 15 '15 at 15:27
• The wikipedia article you linked gives "Kloeden, P.E., & Platen, E.: Numerical Solution of Stochastic Differential Equations, Springer, 1992.", but also says that the general RK-method cannot be adapted as it is possible with the Euler method – martini Dec 15 '15 at 15:29
• Yet Kloeden and Platen (I just opened my physical copy) devote section 12.3 in their book to something called "Implicit Strong Runge-Kutta Schemes". They do remark however that these schemes "are not simply heuristic adaptions to stochastic differential equations of the deterministic Runge-Kutta schemes." Instead they are finite-difference approximations to so called Taylor schemes, which might be worth having a look at. (Quick disclaimer: I've only taken courses on the theoretical parts of stochastic calculus as of yet, so I'm no expert in the numerics). – A.Sh Dec 15 '15 at 15:39
• …and of course, Taylor schemes of various orders are indeed presented and studied in Kloeden and Platen. – A.Sh Dec 15 '15 at 15:56
• I will check the book. – gerd Dec 16 '15 at 10:56