Exercise: Verifying the Martingale Property from First Principles

Prerequisites: Martingales (Discrete Time), Conditional Expectation

Problem

Let $X_1, X_2, \ldots$ be i.i.d. random variables with $\mathbb{P}(X_i = +1) = p$ and $\mathbb{P}(X_i = -1) = 1 - p$, and let $S_n = X_1 + \cdots + X_n$ with $\mathcal{F}_n = \sigma(X_1, \ldots, X_n)$.

1. For $p = 1/2$, verify $\mathbb{E}[S_{n+1} \mid \mathcal{F}_n] = S_n$ from the definition of conditional expectation. Conclude $(S_n)$ is a martingale.

2. For $p \ne 1/2$, compute $\mathbb{E}[S_{n+1} \mid \mathcal{F}_n]$ and identify $(S_n)$ as a sub-, super-, or non-martingale.

3. For general $p \in (0, 1)$, consider the process $M_n = \left(\frac{1-p}{p}\right)^{S_n}$. Show that $(M_n)$ is a martingale. (This is the exponential martingale for a biased random walk — it plays a key role in the gambler's-ruin problem.)

4. Confirm part 3 by direct calculation of $\mathbb{E}[M_{n+1} \mid \mathcal{F}_n]$, showing each intermediate step. Which algebraic identity makes the cross-term vanish?

Hint

For part 3, factor $M_{n+1} = M_n\cdot q^{X_{n+1}}$ where $q = (1-p)/p$, and compute $\mathbb{E}[q^{X_{n+1}}]$ directly.