this is my first post, please excuse my non-technical language.
I would like to simulate the return of a stock, that is correlated with other stocks, that meets the mean and variance of a given empirical time series. The usual way to do this is to set up a geometric brownian motion and to use a cholesky factorization on the correlation matrix of the returns of a set of stocks. As far as i know the moments of this GBM are functions of time but the log of the GBM doesn't contain any skewness.
My question is: In what way can i extend the GBM to introduce a target level of skewness while keeping the other moments as in the standard GBM? Would you mind providing the explicit form of the SDE?
Thank you very much in advance! Thomas