Ito process with white noise

Question

I would like to solve following system of SDEs,

dx_i[t] = f[x_i[t]]*dt + dw_i[t]

where d denotes \[DifferentialD] and w_i[t] corresponds to white noise, which means that $\langle w_i(t)\rangle = 0$ and $\langle w_i(t)w_j(t')\rangle = R\delta_{ij}\delta(t-t')$ ($\langle...\rangle$ denotes time averagin). I know that there is WhiteNoiseProcess function in Mathematica but when I try

w_i ~ WhiteNoiseProcess[1]

my code do not work. So, I technically write down

ItoProces[dx_i[t]==f[x_i[t]]*dt + dw_i[t], w_i ~ WhiteNoiseProcess[1],...]

Naively, I tried WienerProcess[0,1] which is obviosly incorrect set-up. So, my questions are:

1. How to solve numerically SDE with white noise in right hand side?
2. Does ItoProcess function is the simplest and most appropriate tool for solving SDEs in Wolfram Mathematica?
3. If ItoProcess is appropriate function for solving coupled system of SDEs, how to implement white noise?

Answer 1

WhiteNoiseProcess is a discrete-time process, so I'm not sure what you intend by using its derivative "dw_i" in your SDE.

Do you mean instead the continuous WienerProcess? If so, see the first example in the documentation for ItoProcess to set up and solve a SDE.

ItoProcess can also handle coupled systems.