Expanding a bit on chrisaycock's answer, and noting in particular from the abstract
In mathematical finance, solutions to obstacle problem for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset.
we can see that this would be used to price those few rare cases of perpetual options.
The only traded examples I know of are perpetual convertible preferred securities, for example from Wells Fargo's offerings. Such securities are lightly traded by the market players and therefore not always analyzed using the full machinery of a stochastic vol model, even if they should be in principle.
In practice, these "perps" are so bond-like that it is often more useful to think of them as fixed-income instruments. The main concern with them is that the issuer will stop paying the dividends or change capital structure, so it is a bit ridiculous to spend one's time on a fancy stochastic vol model when all the interesting stochastic events have to do with unrelated variables such as alterations in capital structure.