# Numerical solution of SDEs with colored noise

I am trying to numerically solve an SDE with both white and colored noise that models a non-linear circuit: $$dX_t = f(X_t) dt + \sigma_w dW + \sigma_c dC$$ where $W$ is a standard Brownian motion and $C$ is a fractional Brownian motion with Hurst exponent zero (corresponding to $1/f$ flicker noise in electronic circuits). I've found several references for the theory of SDEs with fractional Brownian motion, but not being a mathematician they are way over my head.

Could someone explain what's the harm in using, say, a strong order 2 method from Kloeden and Platen's book and just ignoring the fact that $C$ is colored?

The gap between fractionary Brownian in the usual sense (with positive $H$) and your $C$ is enormous: $C$ is a random distribution (and moreover defined up to a constant, which is harmless if only $dC$ is used), for which "$f(C_t)$" for instance doesn't make any sense. For this reason I think your problem is not just to approximate a well-defined object. The object itself (your process $X_t$) doesn't exist at all.

A numerical method for an SDE with colored noise should work with fractionary Brownian:$$dX_t=f(X_t)+dB^H_t, \ 0<H<1$$but you should not expect it to catch anything meaningful in the limiting case $H=0$.

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.

• The link still doesn't work. Also, what's about the existence of solutions in such a case? The Lipschitz condition is not satisfied Commented Feb 15, 2021 at 21:33

This is a common first step. But the problem is that you can't assure that such method will converge. Because of this, you have to find a way to justify that your method gives "good" approximations.

• Yes, when I just tried it to see what happened, it was pretty disasterous. I ended up just dropping the colored noise source since, ahem, I'm not a mathematician... Commented Jul 25, 2015 at 3:16

This is called mixed SDEs, usually Platen's scheme does not work for it. Euler method is working for it. Another way is , try to mix dW and dC if possible and produce a new noise with a specific correlation (Color noise) to avoid mixed equation. after mixing it will not be a mixed equation , it will be an equation with color noise. then you can apply Platen's RK method. you can generate color noise with circulant embedding method. Hope it would be helpful.