# Poisson Martingales and Gambler's Ruin

1. Feb 6, 2010

### RedZone2k2

1. The problem statement, all variables and given/known data
Find the probability of an outcome of Gambler's Ruin using a Poisson martingale.

Let m be the parameter of a Poisson Process (ie the lambda)
Let N(t) be a continuous Poisson process at time t>=0
Let M(t) = N(t) - mt

2. Relevant equations
Now, define s = inf{t | M(t) <= -a or M(t) >= b} (ie the stopping time)
Let r = P(M(s)>=b)

3. The attempt at a solution
I have shown that M(t) is a martingale with E[M(t)] = 0 for all t
Using the martingale principle, I have:
E[M(t)] = 0 = a*(1-r) + b*r
r = a/(a+b).

This is the solution for discrete gambler's ruin. However, my professor says that this value should be an upper bound for r, and the lower bound should be a/(a+b+1) <= r. I'm not sure how to get this.

My guess should be that it has something to do with the Poisson distribution, and the only difference is that b is bigger by 1. But, if there is a Poisson jump at some episilon of time, its just as if we have a+1 dollars, and our opponent has b-1 dollars; hence I should see a change in the numerator of the bound (and not the denominator). Is there any intuition for this? I'm not necessarily just looking for the answer (or equation), but more of understanding why there should be two bounds to my probability.

Thanks!