My objective is to show the distribution of a portfolio's expected utilities via random sampling.
The utility function has two random components. The first component is an expected return vector which is shocked by a random Gaussian variable ($ER$). The shock shifts the location parameter of the return distribution.
The second random component is a covariance matrix which is multiplied by a random scalar $S$. The scalar is drawn from a normal random variable, centered on $1$, where the variance of $S$ corresponds to our uncertainty in whether volatility increases or decreases.
My initial plan was to take separate draws of the $S$ and $EV$ and simply calculate the utility. However, clearly these random variables are not conditionally independent. In particular, research suggests that when volatility is increasing (i.e. $S > 1$), the expected returns distribution has a lower overall mean. Or, if volatility is contracting rapidly it is likely that the expected returns distribution has a higher mean.
In other words, I should be sampling from the joint distribution of $EV$ and $S$ as opposed to the marginal distributions of $EV$ and $S$ separately.
What's a good technique to estimate and sample these random variables from their joint distribution? The "correct" approach I can think of involves defining a set of states, estimating the transition probabilities between state pairs, and sampling $EV$ and $S$ conditional on random draws of a third state variable. Seems like overkill!
A crude variation of this would be to build a transition matrix such as [ High Vol to High Vol, Low to Low, Low to High Vol, High to Low Vol ] where the location and scale parameters ($ER$ and $S$) are informed by an "expert" (i.e. casual inspection of the data).
Are there other techniques that I may be missing that provide a solid "80/20 rule" solution for sampling from this joint distribution, or is state-space (markov models and the like) the only way to go? For example, perhaps there is a non-parametric technique to estimate the relationship between these two variables.