Martingales and Optional Stopping · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Fair games and when you may stop

Mathematics · Probability · ages 23-24
Name ______________________   Date ____________

  1. A process with expected next value given the past equal to the current value is a martingale.

    Circle one:   True   False

  2. Which of these processes is a martingale?

    • A walk with upward drift 0.1 per step
    • The squared walk S_n squared
    • The symmetric simple random walk
    • The absolute walk |S_n|

  3. A process satisfies E[S_{n+1} given the past} = S_n at every step. A student calls it a martingale. Is that right?

    Circle one:   True   False

  4. A simple symmetric walk starts at 0. What is its expected position after 3 steps?

    Answer: ______________

  5. A student says optional stopping applies to every stopping rule for a martingale. Is that right?

    Circle one:   True   False

  6. Which stopping rule can break the hypotheses of optional stopping?

    • Stop when fortune first hits +10, with no cap on time
    • Stop after exactly 5 bets
    • Stop at the earlier of ruin and time 100
    • Stop when the fortune first leaves [-5, 5]

  7. A gambler with 3 coins bets 1 per fair coin flip, stopping at 0 or 10. What is the chance of reaching 10?

    Answer: ______________

  8. Which stopping rule breaks the theorem?

    • Stop after a fixed number of fair flips
    • Stop at a preset target or ruin bound
    • Stop at the peak of the finished path

  9. A player doubles stakes without limit until the first win and claims a sure profit with fair means. What is wrong?

    • The unbounded stakes break boundedness, so the mean is not preserved
    • The coin used has the wrong denomination
    • The target profit was set too low

  10. A stopping time is bounded and each stopped variable is integrable. A student says optional stopping applies. Is that right?

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Fair games and when you may stop W1-mt_198uXX3jRJ-s1

  1. True · That conditional mean identity is the martingale property itself.
  2. The symmetric simple random walk · Only the symmetric simple random walk has zero-mean increments given the past; drift, squares, and absolute values all trend.
  3. True · That conditional mean identity is the martingale property itself.
  4. 0 · Symmetry keeps the mean at the start.
  5. False · Unbounded or heavy-tailed rules can break the conclusion even for fair games.
  6. Stop when fortune first hits +10, with no cap on time · Unbounded waiting with no cap breaks integrability, so stopping when fortune first hits +10 with no cap is the rule that breaks the hypotheses.
  7. 0.3 · For simple symmetric ruin from k to N, the chance is k/N = 3/10 = 0.3.
  8. Stop at the peak of the finished path · Peeking at the whole path is looking ahead.
  9. The unbounded stakes break boundedness, so the mean is not preserved · Unbounded doubling hides a rare catastrophic loss.
  10. True · Bounded horizons plus integrability satisfy every hypothesis.
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