optional stopping in continuous time
A stopping time T is a random time decidable from the past: {T <= t} is in F_t for every t (in continuous time one uses <= rather than < precisely because of right-continuity). Optional stopping is the assertion that a martingale stays fair when sampled at stopping times: E[M_T] = E[M_0], and more generally M is still a martingale along stopping times, E[M_T given F_S] = M_S for stopping times S <= T. It is the single most-used tool for computing with martingales — hitting probabilities, expected exit times, gambler's-ruin-type identities — and the continuous-time version carries genuine hypotheses that cannot be dropped.
The precise statement: let (M_t) be a cadlag martingale and S <= T stopping times. If T is bounded (T <= c a.s.), then E[M_T given F_S] = M_S. The boundedness of T is essential; to allow unbounded T one needs an integrability upgrade, and the cleanest is uniform integrability: if M is a UI martingale (equivalently, closed by some integrable X, so M_t = E[X given F_t]), then E[M_T given F_S] = M_S for ALL stopping times S <= T, including T = infinity where M_infinity is the closure. Other sufficient conditions for unbounded T include: M is dominated by an integrable random variable, or M_{T and t} is uniformly integrable in t. The continuous-time proof typically discretizes T by dyadic approximation from above (so the approximations stay stopping times, by right-continuity) and passes to the limit using the cadlag paths and Doob's convergence theorems.
Optional stopping is where honesty matters most, because the naive E[M_T] = E[M_0] is FALSE without a hypothesis. Brownian motion is a martingale, yet if T is the first hitting time of level 1 then M_T = 1 always, so E[M_T] = 1 != 0 = E[M_0] — T is finite a.s. but unbounded and B is not UI. The exponential martingale exp(B_t - t/2) stopped never converges to a closing variable. The remedy is always to verify one of: bounded T, uniform integrability, domination, or L^p control. Pairing optional stopping with a cleverly chosen martingale (and often with the quadratic-variation martingale M_t^2 - <M>_t) is the standard route to exit distributions and Wald-type identities.
For Brownian motion started at x in (0,1), let T be the exit time of (0,1). Since B is a continuous martingale and T is a.s. finite with B_{T and t} bounded (UI), optional stopping gives x = E[B_0] = E[B_T] = 1·P(hit 1 first) + 0·P(hit 0 first), so the probability of reaching 1 before 0 is exactly x.
Bounded-before-exit makes B_{T and t} uniformly integrable, which is exactly what legitimizes optional stopping at the unbounded exit time.
E[M_T] = E[M_0] can be flatly false: for Brownian motion and T = first hit of 1, E[M_T] = 1 not 0 — always check boundedness, uniform integrability, or domination before applying it.