I don't know, how to find the probability analytically, but numerical simulation is actually simple. It suggests, that probability the process will remain remains finite is equal to with probability $0.162\pm 0.003$. Hereare the there is a simulation in Wolfram Mathematica 7.0.1. Typical examples of processes with 20+ steps (are shown below. 20 steps are drawn, initial . Initial triangle is always in the center, but on some pictures it is too small to see it): it.
I don't know, how to find the probability analytically, but numerical simulation is actually simple. It suggests, that probability the process will remain finite is equal to $0.162\pm 0.003$. Here are the simulation in Wolfram Mathematica 7.0.1. Typical examples of processes with 20+ steps (20 steps are drawn, initial triangle is always in center, but on some pictures it is too small to see it):