Non-diagonal noise correlation matrix

Hi all!

I am newbie to julia and I am playing with it to solve SDE systems, numerically.
I had a look at Non-diagonal noise Tutorial (Example No. 4), but still something is not clear for me which I explain below:

Lets assume that we have two coupled SDEs

dx1 = a11x1dt + a12x2dt + dw1
dx2 = a21x1dt + a22x2dt + dw2
a_ij’s here are all state dependent. dw1 and dw2 have Non-diagonal, state dependent (x1 and x2 dependent) cross-correlation so that we can generally say:

dw_i*dw_i = b_ij dt

where b_ij is the element-wise representation of noise correlation matrix:

How should one solve this sort of problem using non-diagonal noise option in julia?
Which solver is recommended?