I would like to simulate a bivariate process where the two components, say $X_1(t) $ and $ X_2(t), $ which are related by the following stochastic differential equations:
dX_1(t) = -X_2(t)dt + \sigma_1dW^{(1)}(t)
and
dX_2(t) = X_1(t)dt + \sigma_2dW^{(2)}(t).
I would also like to allow the two BMs $W^{(1)} $ and $W^{(2)} $ that are correlated.
What would be the proper way to do this by defining a process using ItoProcess first and then simulating it via RandomFunction? If the coefficients were just a function of $t $ and not of $X_i(t, $ then I am able to make it work, but everytime I try to enter the process as described in the two SDEs above, I get some error message to the tone that "process" does not exist.
Any help much appreciated. I tried to make sense of what Mathematica has for explanation in the online help, but it does not seem to solve my issues.
Thank you.
Maurice