I would like to simulate the solution of the following system of SDE's: \begin{equation*}\begin{split} dX(t) &= - \frac{1}{2}X(t)\, dt - Y(t)\, dW(t)\\ dY(t) &= -\frac{1}{2}Y(t)\, dt + X(t)\, dW(t). \end{split}\end{equation*} I am using Mathematica for this purpose and in particular the built-in function ItoProcess. I can model each equation separately. But what is the syntax to write both equations so that they use the same brownian motion $W(t) $ ? I checked the documentation in Mathematica, but I cannot quite make out the form to use. Also, once the simulation is done, the result is reported as TemporalData. I would like a plot of (X(t), Y(t)) and not separate plots of $(X(t), t) $ and $(Y(t), t). $ Is there a way to do that?
Sorry if the question is not too challenging, but I tried to use what I know for bi- or tri-dimensional Wiener processes in Mathematica and no success as I assume that ItoProcess works somewhat differently.
Thank you in advance for any help.
Maurice
After some trials and errors.. I think I solved my own question. The program allows also to use two different Brownian motions. In my case I wanted the same to be used to produce the motion on a circle. I also added a graphic of the result I got. The parameters should be easy to modify.
Maurice