I'm trying to figure out if stochastic calculus is the right approach for this problem... but I only vaguely understand it and I am trying to gauge if I need to spend the time learning measure theory etc etc for something that could be much more trivial.

I am going to make some simplifications here (e.g. normal instead of lognormal), so please bear with me. I'm just trying to get my bearings.

Let's say I have two stocks, X and Y.

Stock X is worth 100 and the price is normally distributed with $\sigma\_X = 1$ per period.

Stock Y is worth 99 and the price is normally distributed with $\sigma\_Y = 3.3333$ per period.

The two stocks are 90% correlated and the beta is $0.9 \cdot 3.333/1 = 3$

So what if I said to you, in one period, I'll give you stock X or stock Y, whichever has a greater value.

So the question is... what's the expected value of the max of the two stocks?

I can, of course, simulate this scenario, but is there a way to get the answer closed form?

Using Cholesky decomposition, I simulate two correlated random normals and give them the appropriate mean and standard deviation.

I approximate the value of $\max(X,Y) \approx 100.6$