One way of dealing with colored noise is to represent it by another SDE. For example, here is a presentation which presents some models like:

$$dx = (1+\frac{1}{2x} - \frac{x}{2\times 10^3})x^{4} dt + x^{\frac{5}{2}} dW_t $$

has a solution with a $1/f$ spectrum. So now you just have to solve a system of nonlinear SDEs.

I wouldn't use the Strong Order 2 Method in practice because it takes so many more computations I don't think it actually gives a speedup. Rossler has some Runge-Kutta type methods which are easier to implement. But honestly try Euler-Maruyama / Milstein first and see if that's good enough for your needs before you do something fancy.

One way of dealing with colored noise is to represent it by another SDE. For example, here is a presentation which presents some models like:

$$dx = (1+\frac{1}{2x} - \frac{x}{2\times 10^3})x^{4} dt + x^{\frac{5}{2}} dW_t $$

has a solution with a $1/f$ spectrum. So now you just have to solve a system of nonlinear SDEs.

I wouldn't use the Strong Order 2 Method in practice because it takes so many more computations I don't think it actually gives a speedup. Rossler has some Runge-Kutta type methods which are easier to implement. But honestly try Euler-Maruyama / Milstein first and see if that's good enough for your needs before you do something fancy.

Update: I came back to this as my own notes and noticed that the link was broken. Here's another paper with an SDE model which gives $1/f^\beta$ spectra.