Optional sampling and uniform integrability

Write a martingale stopped at T as its initial value plus its increments multiplied by the indicators that T has not occurred before the corresponding step. Those indicators are predictable, so each retained increment has mean zero. If T is bounded by a fixed integer N, the finite sum is integrable and taking expectations gives the original mean. More generally, for stopping times S at most T at most N, conditional expectation of the stopped terminal value given the stopped information at S equals the value at S.

The stopped sigma-algebra consists of events whose intersection with stopping by each time is measurable by that time. It records what is known when the rule actually stops, rather than information at one predetermined index. The bounded conditional sampling statement can be checked by testing these events and splitting them over the possible stopping values. This formulation makes explicit both the information constraint on the times and the integrability needed for the conditional expectation to exist.

For an almost-surely finite T, the stopped variables converge almost surely to the value at T, but their expectations need not converge. A sufficient additional condition is uniform integrability of the family of stopped variables indexed by deterministic n. This gives convergence in L1, so the bounded expectation identities pass to the limit. Bounded stopped values provide an easy sufficient condition. Another sufficient setting is a martingale closed by one integrable terminal variable: each value is its conditional expectation under the current information.

A closed martingale is uniformly integrable, and at finite stopping times it remains the conditional expectation of that same closing variable given the stopped information. This yields conditional optional sampling between almost-surely finite stopping times. Other hypotheses are possible, such as bounded increments together with an integrable stopping time for the expectation identity, but they cannot be silently exchanged for almost-sure finiteness. The applicable sufficient condition should be named before removing a deterministic truncation.

For a fair nearest-neighbor walk stopped upon reaching either of two finite barriers, the stopped positions stay between those barriers. Absorption occurs almost surely: from any interior site, a fixed finite sequence of steps has a positive probability of reaching a barrier, giving a geometric bound on survival over blocks. Thus bounded positions justify optional sampling for the walk. The terminal values then convert the preserved mean into a hitting probability. No independence of the stopping time and the walk is required.

To compute the expected duration, apply the bounded theorem to the squared walk minus time at the truncated stopping time. The squared stopped positions are uniformly bounded and converge to the squared terminal position. The truncated times increase to the stopping time, so monotone convergence identifies their expected limit. This proves the duration is integrable instead of assuming that fact to justify the very identity used to calculate it. A one-sided barrier has no analogous uniform bound on stopped positions and requires a separate argument.