I know that it is possible to simulate two correlated GBM in e.g. Matlab (Generating Correlated Asset Paths in MATLAB) based on cholesky decomposition. However, they take as input the correlation matrix, which from my understanding is just the Pearson correlation coefficient. However, if I look at correlation between two time series, cross-correlation is the correct measure. This also results in a lag. Is it possible to incorporate the lag into the simulation of the correlated GBM?