Consider the gambler's ruin problem, where a player plays a...

Consider the gambler's ruin problem, where a player plays a sequence of independent games, either winning one dollar with probability $1 / 2$ or losing one dollar with probability $1 / 2$. The player continues until either losing $\ell_{1}$ dollars or winning $\ell_{2}$ dollars. Let $X_{n}$ be 1 if the player wins the $n$th game and $-1$ otherwise. Let $Z_{n}=$ $\left(\sum_{i=1}^{n} X_{i}\right)^{2}-n$. (a) Show that $Z_{1}, Z_{2}, \ldots$ is a martingale. (b) Let $T$ be the stopping time when the player finishes playing. Determine $\mathbf{E}\left[Z_{T}\right]$. (c) Calculate $\mathbf{E}[T]$. (Hint: You can use what you already know about the probability that the player wins.

Consider the gambler's ruin problem, where a player plays a sequence of independent games, either winning one dollar with probability $1 / 2$ or losing one dollar with probability $1 / 2$. The player continues until either losing $\ell_{1}$ dollars or winning $\ell_{2}$ dollars. Let $X_{n}$ be 1 if the player wins the $n$th game and $-1$ otherwise. Let $Z_{n}=$ $\left(\sum_{i=1}^{n} X_{i}\right)^{2}-n$.
(a) Show that $Z_{1}, Z_{2}, \ldots$ is a martingale.
(b) Let $T$ be the stopping time when the player finishes playing. Determine $\mathbf{E}\left[Z_{T}\right]$.
(c) Calculate $\mathbf{E}[T]$. (Hint: You can use what you already know about the probability that the player wins.

Key Concepts

Gambler's Ruin Problem

The gambler's ruin problem is a classical problem in probability theory that studies the fate of a gambler who wins or loses a fixed amount repeatedly until reaching a predetermined gain or loss limit. It illustrates the behavior of random walks with absorbing boundaries and helps in understanding the probability of eventual ruin or success along with metrics such as expected duration of play.

Optional Stopping Theorem

The optional stopping theorem provides conditions under which the expected value of a martingale at a stopping time is equal to its initial value. This theorem is crucial in problems like the gambler's ruin, where it allows the computation of expected values (such as the expected time to termination) by considering the martingale at the stopping time.

Martingale

A martingale is a type of stochastic process in which the conditional expected future value, given all past information, is equal to the present value. This concept formalizes the idea of a “fair game,” where there is no advantage or predictable gain over time. The process Z? in the problem is shown to be a martingale by verifying this key property for each step.

Stopping Time

A stopping time is a random variable that represents the time at which a given stochastic process meets a particular criterion, with the decision to stop being based only on past and current information. In the context of the gambler’s ruin problem, the stopping time is the moment when the game ends because either a prescribed winning or losing threshold has been reached.

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