Any short cut way of American perpetual option in finite difference, I asked several finite difference Gurus about this and nobody had a elegant solution on top of their head. For standard perpetual American we have close form...but I am thinking about finite difference, reason is if you suddenly want some type of different pay off etc...then a numerical solution is often preferable, but a good start should be to value plain American perpetual option, because we can check it against benchmark (closed form)...or also if you want to test out a new closed form agianst numerical method to see if about same result...One simple approximation of course use very long time to maturity in grid, but this is far from elegant method. Some type of elegant short cut should be possible here as option always have same time to maturity =infinite....would be very nice to solve it by for example single time step in grid or something like that, but not as easy as it seams ?

What about doing a coordinate transform in T that transforms T=infinity to some finite value?

I guess d/dt disappears from the eq. and it becomes an elliptic eq. (in case of of 1-d BS it becomes an ODE; better to say elliptic ineq. if you mean american vanilla option, I am not sure here). For 1-d elliptic eq. (ODE), I suppose analytical solutions should be easy to get. Aside from that, among numerous schemes for elliptic eq. some of them are iterative and are equivalent to solving heat eq. with T going to infinity.