Section 16.5

Martingales and Optional Stopping

Constructing martingales (exponential and product martingales) and using optional stopping to obtain probabilities or expected stopping times.

Toolkit

A process $M_n$ is a martingale with respect to $\mathcal{F}_n$ if $\mathbb{E}[M_{n+1}\mid\mathcal{F}_n]=M_n$ . Given i.i.d. steps $X_i$ with moment generating function $\phi(\theta)=\mathbb{E}[e^{\theta X}]$ , the exponential martingale is

M_n=\frac{e^{\theta S_n}}{\phi(\theta)^n},\qquad S_n=\sum_{i=1}^n X_i,

since $\mathbb{E}[e^{\theta X_{n+1}}\mid\mathcal{F}_n]=\phi(\theta)$ makes the ratio’s conditional mean exactly $1$ . More generally, any product $M_n=\prod_{i=1}^n Y_i$ of independent positive factors with $\mathbb{E}[Y_i]=1$ is a martingale, and $M_n^p$ is a martingale iff $\mathbb{E}[Y^p]=1$ ; by convexity of $p\mapsto\mathbb{E}[Y^p]$ this typically holds at $p=1$ and one other root $p^*\neq1$ .

Optional stopping theorem: if $T$ is a stopping time satisfying suitable boundedness/integrability conditions, $\mathbb{E}[M_T]=\mathbb{E}[M_0]$ . Apply with the exponential martingale evaluated at a root $\theta^*$ making $\phi(\theta^*)=1$ to extract hitting probabilities, or differentiate in $\theta$ at $\theta=0$ (Wald’s identities) to get $\mathbb{E}[T]$ and $\operatorname{Var}(S_T)$ .

Problems