Poisson Martingales and Gambler's Ruin

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In summary, using a Poisson martingale, we can find the probability of an outcome of Gambler's Ruin. The upper bound for this probability is a/(a+b), assuming each Poisson jump is a win or a loss. The lower bound takes into account the possibility of a tie and is given by a/(a+b+1). This is because the Poisson distribution has a small chance of no change in the gambler's money, reducing the probability of reaching the boundaries a and b.
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Homework Statement


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

Homework Equations


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

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!
 
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Thank you for your question. The concept of Gambler's Ruin is a classic problem in probability theory and has been studied extensively. It is interesting that you are approaching it using a Poisson martingale, which is a valid approach.

To understand why there are two bounds for the probability, we need to consider the properties of the Poisson distribution. The Poisson distribution is a discrete probability distribution that describes the number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independently of the time since the last event.

In the context of Gambler's Ruin, we can think of the Poisson jumps as representing the wins and losses of the gambler. The parameter m represents the average rate of these wins and losses. Now, let's consider the two bounds for the probability:

1. Upper bound: r = a/(a+b)
This is the solution for discrete Gambler's Ruin, as you have correctly stated. It assumes that each Poisson jump is either a win or a loss, with equal probability. This is a valid assumption for a discrete process, but in reality, the gambler can also experience a tie (no change in the amount of money). This means that the probability of reaching the boundaries a and b is slightly lower than a/(a+b). Hence, this is an upper bound for the probability.

2. Lower bound: r = a/(a+b+1)
This bound takes into account the possibility of a tie, which was not considered in the upper bound. It assumes that there is a slight chance of no change in the gambler's money, which slightly reduces the probability of reaching the boundaries a and b. This is a more accurate representation of the probability and hence, is a lower bound.

I hope this explanation helps you understand the two bounds for the probability. Good luck with your further studies!
 

What is a Poisson martingale?

A Poisson martingale is a mathematical concept used in probability theory and statistics. It is a process that follows a Poisson distribution and has the property of being a martingale, meaning that the expected value of the process at a future time is equal to its current value. In other words, a Poisson martingale is a stochastic process that has no predictable trend and its expected value remains constant over time.

How is a Poisson martingale related to Gambler's Ruin?

Poisson martingales are closely related to Gambler's Ruin, a classic problem in probability theory. In Gambler's Ruin, a gambler with a finite amount of money plays a game against a casino with the goal of winning a specific amount. The gambler's wealth can be modeled as a Poisson martingale, and the problem is to determine the probability of the gambler's ruin, or the probability that they will lose all their money before reaching their desired goal.

What are some real-world applications of Poisson martingales and Gambler's Ruin?

Poisson martingales and Gambler's Ruin have numerous applications in various fields, including finance, insurance, and gambling. In finance, they can be used to model stock prices and predict future market trends. In insurance, they can be used to assess risk and determine appropriate premiums. And in gambling, they can be used to analyze and improve betting strategies.

How can Poisson martingales and Gambler's Ruin be solved?

There are several methods for solving problems involving Poisson martingales and Gambler's Ruin. One common approach is to use generating functions, which allow for the calculation of probabilities and expected values. Another method is to use recurrence relations, where the probabilities of different outcomes are expressed in terms of each other. Additionally, simulations and numerical methods can also be used to solve these problems.

What are some limitations or assumptions of using Poisson martingales and Gambler's Ruin?

Like any mathematical model, Poisson martingales and Gambler's Ruin have certain limitations and assumptions. One major assumption is that the processes are independent and have no external influences. In reality, external factors such as market trends or human behavior can impact the outcomes. Additionally, these models assume a perfect knowledge of the underlying processes, which may not always be accurate in real-world scenarios.

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