Confine the random walk of the previous two notes between two absorbing barriers — a floor at and a ceiling at — and a new pair of questions becomes unavoidable: which barrier does the walk hit first, and how long does that take? This is the gambler's ruin problem, the oldest worked example in the theory of Markov chains (it dates to a 1656 letter between Pascal and Fermat), and it is also the first place this unit uses first-step analysis: condition on the outcome of the very next step, and turn a probability question about an entire infinite-horizon process into a finite difference equation. First-step analysis is the single most important proof technique in this unit — it reappears, essentially unchanged, for general Markov chains, continuous-time chains, and branching processes.
Learning objectives. By the end of this note you should be able to:
- set up the gambler's ruin chain and verify it has the Markov property;
- derive the first-step difference equation for the absorption probability and state its boundary conditions;
- state (without full derivation — see the companion derivation note) the closed-form ruin probability and expected duration;
- connect gambler's ruin to the binomial-tree boundary conditions used in option pricing.
In this lesson
- The gambler's ruin chain
- First-step analysis
- Duration of the game
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