Are there any techniques that can make a multivariate random number generating process for stock prices/returns, like geometric Brownian motion via Cholesky, also include the simulation of a finite number of market regimes (say 2 or 3) so that systematic (market-induced) movement in prices are experienced across all assets for the same span of time intervals? (e.g. observations/prices 1-250 are market regime 1 for all assets, prices 251-400 are regime 2, etc)
For the univariate case, I understand that simulated returns can be generated from separate Gaussian distributions, each of which represents a "bullish" or "bearish" market regime, with:
- the returns for the bullish regime drawn from a Guassian distribution with positive mean and low variance,
- while returns for the bearish regime draw from a Gaussian distribution with slight negative mean but higher variance,
but my question pertains to generating multivariate artificial returns instead of one-by-one univariate.