@Hassan before you delve into that book (ST94), let me reiterate that it considers stable processes generally. What many people refer to as THE stable process is called alpha-stable Levy motion in ST94. That and stable subordinators might possibly be the only Levy processes they consider in their path property analysis (chapters 9-12).

hmmm... i don't know of anything. I would try discretizing by using a random walk where each step is a small Gaussian, and using an approximation of the delta distribution. A nice approximation to the delta function is again a Gaussian with very small variance.