Cuch, For me I think of pencil-and-paper solutions of PDEs as being really, really easy to screw up. But that's just me. Think of the original solution to Black-Scholes. There are a thousand ways to goof up on the way toward transforming into the heat equation.
That's interesting because I would say that there are far more ways to goof up other methods!
Anyway, unless it's an exam question you might not have to transform anything.
If your problem is novel then you should use whatever tools are applicable. Without prejudice!
I find the following with martingale methods:
1. If the problem is trivial then you can use them.
2. If the problem is non trivial then you usually end up with "...and then you solve by Monte Carlo simulation." So why bother with all that heavy machinery if the destination is always the same?
And the second case is usually only when you have the possibility of hedging. Which is pretty much all that martingalists ever consider!