Martingales and stopping times

A filtration is an increasing family of sigma algebras representing information revealed through time. A martingale Mₙ is adapted, so its current value is known at time n; each value is integrable; and its next conditional expectation equals its present value. A symmetric random walk with independent increments of plus or minus one is the basic example. Fairness refers to expected increments given the past, not to the path remaining close to zero or alternating wins and losses.

The filtration matters. Revealing tomorrow's increment today would destroy the martingale property of a nontrivial symmetric walk relative to that enlarged information. Conversely, an integrable terminal payoff Z generates a martingale Mₙ=E[Z|Fₙ] by the tower property. This is a model for updating predictions as evidence arrives: realized predictions change, but their conditional future mean equals the current prediction. Zero unconditional mean increments alone do not establish this stronger information-dependent property.

A nonnegative integer-valued random time T is a stopping time when the event T≤n is determined by information available at n, for every n. The first time a walk hits a threshold qualifies: by time n, the observed path reveals whether a hit has occurred. The last time a walk visits a threshold before an unknown future endpoint generally does not qualify, because deciding that a visit was the last one requires inspecting future observations.

For a bounded stopping time and an integrable discrete-time martingale, optional stopping gives E[M_T]=E[M₀]. For unbounded stopping times, further hypotheses are essential. One common route is uniform integrability of the stopped family; another uses bounded increments together with finite expected stopping time. Merely knowing T is finite almost surely is insufficient. Always identify the theorem version and show its hypotheses rather than appending an informal phrase about waiting until the favorable event eventually occurs.

An unbounded stopping time can be truncated to T∧n. Each truncated time is bounded, so the corresponding expectation identity is valid. Passing n to infinity requires control of the stopped variables. Bounded convergence works when their magnitudes share a deterministic bound; uniform integrability is a more general tool. Almost-sure convergence alone does not preserve expectations, because probability mass can escape through rarer and larger values. This is the same obstruction that appears in disappearing-spike examples from integration theory.

For a symmetric walk started at zero, the first hit of +1 is finite almost surely, and the stopped value is exactly one. If unrestricted optional stopping applied, its expectation would simultaneously be zero and one. The contradiction identifies a failed hypothesis: this one-sided hitting time has infinite mean, and the truncated stopped walk is not uniformly integrable. In contrast, a walk stopped on leaving a finite interval stays bounded, allowing a controlled gambler's-ruin probability calculation. Keeping these two exit rules side by side helps isolate the hypothesis: both eventually stop, but only the bounded-interval stopped values admit the simple bounded-convergence argument.