I am attempting to simulate an sde system of the following form $$ dX_t=\sqrt{\vert aX+bY\vert}dW^1_t \\ dY_t=\sqrt{\vert cX+dY \vert}dW^2_t $$ where $W=(W^1,W^2)$ is a standard two dimensional Brownian motion by using the Euler scheme. $$ X(t_{i+1})=X(t_i)+\sqrt{\vert aX(t_i)+bY(t_i)\vert} (W^{(1)}(t_{i+1})-W^{(1)}(t_i))\\ Y(t_{i+1})=Y(t_i)+\sqrt{\vert cX(t_i)+dY(t_i)\vert} (W^{(2)}(t_{i+1})-W^{(2)}(t_i)) $$
. In order to implement an euler scheme in Java I intend to use a Standard normal sample generator from
My question is that every time step of the Euler-Maruyama scheme can i just use normal distribution with variance $t_{i+1}-t_i$ to sample both $(W^{(1)}(t_{i+1})-W^{(1)}(t_i))$ and $(W^{(2)}(t_{i+1})-W^{(2)}(t_i))$.
How can I ensure that the two brownian motions generated are actually independent . Do I need a clever trick in order to implement this or the pseudorandom generator generates independent sample every single time.
Ofocurse since the coefffients are non-Lipchitz there might be problems of non-convergence but I am not really worried about them since I just use this as an example. I assume that there exists a strong unique solution to this system.