I'm looking for a textbook/journal article reference for the well-known result that an increase in volatility increases the value/price of a standard American (call and put) option. In the case of continuous time with geometric brownian motion dynamics, I know how to prove the result directly from standard comparative-statics on the value function; however, I'm looking for a formal reference as every time I read something related on an article it is taken as given with no formal reference to the fact or only numerical examples are provided.
I don’t think that you will ever find a “demonstration” of it, for two reasons. First, as you mentioned, it can be obviously deduced from comparison arguments: American option price is always no less than European option price, which monotonically increases with volatility. Second, because there are no analytic framework for American option. Their valuation is a dynamic programming problem that pretty always needs to be solved numerically (either by finite differences methods or by a smart Monte Carlo). If you want to convince yourself of that result, just take the “comparison argument with European options”, should be enough.