Let x0=sigma/2.

With probability 1/2, x_(k+1)=min(x/delta, sigma), and
with probability 1/2, x_(k+1)=max((x-bet)/delta, 0)

In other words, if you think of the interval from <0,sigma>, the process I am thinking of starts at the midpoint of the interval. With probability 1/2 we move "up" i.e. to a value x/delta. If this value lies outside of the interval, we move to the highest value sigma, and stay there forever. With probability 1/2 we move down, i.e. to (x-bet)/delta, unless this moves us to a value outside the interval (i.e. a negative value), in which case we move to zero and stay there forever. (It is simple to check that for values in the interval a move "up" increases the value of x while a move down decreases the value of x).

It'seems intuitive that if we let this process run forever, with probability one we will arrive at one of the two absorbing states i.e. zero or sigma. How can I prove this? Also, I would like to be able to calculate the probability that in the long run the process will terminate in state 0 or state sigma. Unfortunately this isn't a simple random walk, because the magnitude of the jumps depends on the current state.

Any advice is much appreciated.

Each of these subintervals has a probability of moving to some of the other intervals or to the endpoints. This creates a descrete Markov process witha matrix of size NxN, with N = sigma/delta, or so. It may be difficult to work with general parameters, but I think that picking up some definite values may give you a good idea of what is happening in general.

Say, for beta = 1 and delta = 2/3, sigma = 3. You could consider 5 intervals [0, 2/3], [2/3, 4/3], [4/3, 6/3], [6/3, 8/3], [8/3, 3]. Call them I₀, I₁, ..., I₄. Now write p(i, k) for the transition probability of moving from I_i to I_k.

Now, numbers below 4/9 remain in the 0th interval with probability 1. Those from 4/9 to 2/3 stay in that interval with probability 1/2 and move to the next interval with probability 1/2. So I would say that

P(0, 0) = 2/3 + 1/3·1/2 = 5/6.

2/3 here is the conditional probability for a point in the 0th interval to be less than 4/9.

p(0, 1) = 1/3·1/2 = 1/6.

And so on. There is a start-up costs but once the matrix has been calculated, all you'll have to do is to consider its powers.

The lengths in the subdivision need not be related to delta. Also, you can add one extra interval below 0 and one above sigma. I think this is a manageable, although, a non-trivial task.