I would like to generate conditional correlated random variables. I have a correlation matrix between normal variables, and these variables are modeled through SDEs.
What are the algorithms to generate such random values? One example I found is; http://www.mathworks.com/help/toolbox/econ/interpolate.html
You can generate correlated gaussian variables by computing the Cholesky decomposition of the variance matrix and multiplying independent gaussian variables with it. There was a similar question on Quant Stackexchange a few days ago.
To simulate the solution of a stochastic differential equation (SDE),
you can just discretize it, i.e., consider it as a difference equation (this is called the "Euler scheme") and use the correlated
brownian motion you want.
In some cases, this is too imprecise,
but it is possible to derive
a second-order approximation of the SDE
from Ito's formula (this is called the "Milstein scheme";
it is an analogue of the Runge-Kutta method).
If you are using R, this is implemented (for instance)
in the sde package
and described in the accompanying book.
