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When using OST in a problem, the workflow is usually:

  1. Choose a martingale $(X_n)$.
  2. Define the stopping time $T$.
  3. Verify an OST condition, usually boundedness before stopping or uniform integrability.
  4. Write
$$ \mathbb E[X_T]=\mathbb E[X_0]. $$
  1. Expand $\mathbb E[X_T]$ using the possible terminal values of $X_T$.

Classic example

For gambler’s ruin with

$$ T=\inf\{t\ge 0:X_t\in\{0,c\}\}, \qquad X_0=a, \qquad 0the simple random walk $(X_n)$ is a martingale and $X_T\in\{0,c\}$.

Thus

$$ \mathbb E[X_T]=a. $$

If $p=\mathbb P(X_T=c)$, then

$$ cp=a, $$

so

$$ p=\frac ac. $$