I don't think I have ever come across the concept of stochastic correlation so I imagine it's not very widespread, but I had the idea to implement a Monte Carlo VaR model for a portfolio of stocks by simulating their price processes using the GARCH(1,1) volatility model and arrived at the question of how to treat the correlations. Is there a way to find correlation parameters by using your calibrated GARCH model? If not would there be anything systematically wrong by just using a correlation matrix calculated by EWMA as you usually would and use that with your stochastic GARCH volatilities to simulate the stock price processes?
As a bonus question, how is correlation treated in general when you're dealing with stochastic volatlity models? Is it the case that stochastic volatility is mostly just used for single stocks when handling derivatives?