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.

  • $\begingroup$ Pseudo-random generators are tested for the correlation of such evenly spaced subsequences, that is, that vectors build from consecutive samples are independent. Up to dimension 20 there should be no problem, the better ones should stay independent for dimensions in the hundreds. $\endgroup$ – LutzL Apr 4 at 13:01
  • $\begingroup$ @LutzL Even if I am simulating a million paths (say with a 100 time discretization points)? $\endgroup$ – user3503589 Apr 4 at 13:22
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    $\begingroup$ Then you should use RNG with a really large cycle length. Widely advertized was the Mersenne twister, but there is valid critique that that algorithm is unnecessarily complicated for the achieved randomness. $\endgroup$ – LutzL Apr 4 at 13:26
  • $\begingroup$ Thank you Mersenne Twister is what I am using. :) $\endgroup$ – user3503589 Apr 4 at 13:37

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