geodesic,
Things get more interesting in 2D and up. I've got one chapter in a recent book that may interest you: Ch. 5 "Stochastic Volatility as a Hidden Markov Model" in the book "Option Valuation under Stochastic Volatility II". This turns out to be kind of a mixed lattice and continuum representation.
The virtue of convergent lattice methods is that they provide an explicit Markov chain approximation to a continuum process -- an approximation that has manifestly non-negative transition probabilities and weak convergence to the continuum process. PDE discretizations typically do not offer this interpretation.
If you've ever tried to solve a PDE for a tricky transition probability in 2D or higher, you've probably seen cases where convergence to a strictly positive result can be quite computationally demanding. And simply truncating a small negative pde result to zero can sometimes be grossly wrong (in maximum likelihood estimation, for example). So, there is a role for both approaches.
I haven't seen that particular problem, but I have seen "bouncing ghosts" where diffusions of probability bounce off the "grid wall" at "infinity". I've also seen the same thing happen off the faux wall at zero in spherical coordinates for a spherically symmetric function (where r is restricted to be non-negative). So yeah, I've definitely seen gotchas in PDE approaches, but I don't think I've ever solved a numerical PDE in finance. Only in physics (my dissertation research was solving and interpreting a complex valued diffusion eqn.)
If you're the author of "Option Valuation under Stochastic Volatility I" may I just say how amazing your book is? I didn't know book 2 existed -- it seems like you covered so much in book 1!