I assume by 'constant coefficients', you actually mean just the diffusion term.
With that, I'd say 'it depends' on whether you want to do a Monte Carlo or a lattice method.
You have to analyze the model to see if S=0 is reachable. If not (and I am assuming S=infinity is not reachable) then the transformation is useful. But if S=0 is reachable, then I'd
- do the transformation anyway for a lattice method, but
- stick with the original coordinates for Monte Carlo.
A simple illustrative case: dS = sqrt(S) dW, where S=0 is reachable. So I will guess you will have Monte Carlo problems near (the Y-image of) small S if you transform it to
dY = b(Y) dt + dW.
On the other hand, if you are doing a lattice method, that coord. transform can be very useful even when the origin reachable.
Also, haven't looked at the paper in a while, but my recollection is they use both the original and transformed coordinates and also allow the binomial process to jump several lattice spacings when needed. So that's a different wrinkle. In other words, you transform to unitized variance, derive some binomial transition rules, and then transform back. So that's for a lattice method. When you do all that, issues with reaching S=0 may be resolved because you are actually making (final) transitions in the original (finite) coordinates.