--- title: "Martingales and Optional Stopping" description: "A martingale is a process whose expected next value, given everything so far, is its current value. Stopping it at a random time preserves that expectation, as long as the stopping rule is not allowed" canonical: https://lightmysky.com/learn/mathematics/martingales-and-optional-stopping-mt_198uXX3jRJ source: https://lightmysky.com/learn/mathematics/martingales-and-optional-stopping-mt_198uXX3jRJ.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Martingales and Optional Stopping A martingale is a process whose expected next value, given everything so far, is its current value. Stopping it at a random time preserves that expectation, as long as the stopping rule is not allowed to look ahead. Subject: Mathematics · Area: Probability · Ages 23 to 24 Page: https://lightmysky.com/learn/mathematics/martingales-and-optional-stopping-mt_198uXX3jRJ ## Ready when they can - Check the martingale property for a given process - Apply optional stopping to compute a ruin probability - Give a stopping rule for which the theorem's hypotheses fail and the conclusion is false ## Lesson: Fair games and when you may stop A martingale is a process whose expected next value, given everything so far, equals its current value. The symmetric walk with steps plus 1 or minus 1 is the model: each step has mean 0, so starting from 0 the mean after 3 steps is still 0. Drifted walks, squares, and absolute values all trend and fail the test. **Example.** Think of a fair coin game priced at zero: neither side profits on average, and repeating the game cannot change that without new information. The binomial coin flips are the laboratory where fairness is first measured. Only the symmetric walk keeps zero-mean increments given the past. Optional stopping says you may stop a martingale at a random time and keep the same expectation, if the rule never looks ahead and stays bounded. A gambler aiming at a target fortune before ruin uses this: the stopped mean still equals the starting fortune, which unlocks the ruin probability. Boundedness, of time or fortune, is the price of the conclusion. **Tip.** A rule that peeks at the future breaks the theorem, and the conclusion can fail badly. Waiting until the peak of a finished path, or doubling stakes without limit until a win, both look ahead or run unbounded. When you audit a stopping rule, check foresight first and boundedness second. **Recap.** Martingales forecast tomorrow as today, fair stopping preserves the mean, and any rule with foresight or no bound voids the promise. ## Practice 18 questions on this page, each with its working shown. ## Needs first - [Conditional Expectation Given a Sigma-Algebra](https://lightmysky.com/learn/mathematics/conditional-expectation-given-a-sigma-algebra-mt_i2sxZc-ER8) - [Expectation and Variance by Integration](https://lightmysky.com/learn/mathematics/expectation-and-variance-by-integration-mt_kRSaKgp3bs) - [Stationary Distributions and Long-Run Behaviour](https://lightmysky.com/learn/mathematics/stationary-distributions-and-long-run-behaviour-mt_MGfHvUGkeh) ## Opens up - [Brownian Motion and Its Defining Properties](https://lightmysky.com/learn/mathematics/brownian-motion-and-its-defining-properties-mt_T5KtWIk07Z)