Martingale and absorption probabilities

Let be the fortune of a gambler at time t. It is assumed that is a discrete time Markov process with state space {0,1,2,...N} where N is a positive integer. For

States 0 and N are absorbing so that

You may assume that is a martingale with respect to itself.

(i) Use the Martingale Convergence Theorem to show that with probability 1 the process is eventually absorbed at 0 or N.

I cannot see how the MCT can show that the process is eventually absorbed at 0 or N as it only shows that for some

Re: Martingale and absorption probabilities

Hello,

Just a quick thought.

Write the definition of a.s. convergence :

Let . We want to show, through the MCT, that for any .

The aim is to prove that by contradiction (I think that's the way...). So let's suppose .

means that .

Assuming that (it's true for any epsilon and we usually choose a small epsilon), it means that , since Xn takes only integer values.

And then try to finish by considering the probability (*) and the fact that . It should be possible to finish the proof (but I've merely written what I was thinking, so I'm not sure...)

Re: Martingale and absorption probabilities

Maybe you can also consider liminf of Xn=0 or N and use Kolmogorov's 0-1 law ?