Simulation of 2-dimensional Brownian motion

Question

I am trying to simulate (for the first time) a 2-dimensional SDE, in Matlab.

$$dX(t)=F(t,X(t))\,dt + \sigma(t,X(t))\,dBt$$

I have no problem using the Euler-Maruyama method in the one dimensional case, but I am not sure if I am correct with the 2-dimensional Brownian motion $B=(B_1,B_2)$.

I need to simulate two sequences $dB_1$ and $dB_2$. Can I just simulate 2 sequences in Matlab $\sqrt{dt}*randn$? Will they be independent ?

I tried to search for tutorials on simulation of two independent Brownian motions. Can you help me with this? What is the easiest way of generating two independent sequences of normally distributed random numbers?

Thank you :) Bilka

Answer 1

You can find the code for a complete 2d simulation here. In any case, this is kind of things where tons of material can be found just using google.

Answer 2

The following code

>> X = cumsum(0.01 * randn(10000,2));
>> plot(X(:,1), X(:,2))
>> axis equal

generates this plot. Adjust parameters to taste.

Answer 3

"Can I just simulate 2 sequences ..."

Yes you can just do it in the straightforward way, and it will do the right thing.