Exercise: Gambler's Ruin for a Biased Random Walk

Prerequisites: Optional Stopping Theorem, Martingales (Discrete Time)

Problem

A gambler starts with $k$ units of capital and plays rounds of a biased game: they win $+1$ with probability $p$ and lose $-1$ with probability $q = 1 - p$, independently each round. They continue until either reaching a goal of $N$ units or being ruined at $0$. Assume $p \ne 1/2$.

Let $S_n$ denote the gambler's wealth and $\tau = \inf\{n : S_n \in \{0, N\}\}$ the exit time. Set $\rho = q/p$.

1. Show that $M_n = \rho^{S_n}$ is a martingale with respect to $\mathcal{F}_n = \sigma(S_0, \ldots, S_n)$.

2. Verify that OST applies: which of conditions (1)–(4) in the theorem is satisfied here?

3. Apply OST to derive the ruin / success probabilities: $\mathbb{P}(S_\tau = N \mid S_0 = k) = \frac{\rho^k - 1}{\rho^N - 1}, \qquad \mathbb{P}(S_\tau = 0 \mid S_0 = k) = \frac{\rho^N - \rho^k}{\rho^N - 1}.$

4. Concrete values. Compute $\mathbb{P}(S_\tau = N)$ for $(p, k, N) = (0.49, 50, 100)$ and $(0.45, 50, 100)$. How does the slight shift in $p$ affect the success probability? Interpret in the context of a casino's house edge.

Hint

For part 1, use independence and compute $\mathbb{E}[\rho^{X_{n+1}}]$. For part 2, note that $M_n$ is trapped between two explicit constants on $\{n \le \tau\}$.