An unbounded stopped exponential martingale needs uniform integrability

Given: AC, (H), an everywhere-continuous zero-start standard Brownian motion $B$ equipped with its usual augmented natural filtration, the process $Z_t=\exp (B_t-t\mathrm{/}2)$, the level $1\mathrm{/}2$, the stopping time $\tau =\inf \{t:Z_t=1\mathrm{/}2\}$, and $t>0$.

Exponential martingale. $Z$ is a positive continuous martingale with $E{Z}_t=1$ for every $t$, and for $\theta =2$ the same statement applied to $\exp (2B_t-2t)$ gives $E{e}^{2B_t}=e^{2t}$, hence $E{Z}_t^2=E{e}^{2B_t-t}=e^t<\mathrm{\infty }$. The exponential Brownian martingale Brownian motion

The level set is hit. On the full-measure event of continuity, $\log Z_t=B_t-t\mathrm{/}2\to -\mathrm{\infty }$ as $t\to \mathrm{\infty }$ because $B_t\mathrm{/}t\to 0$ almost surely by the law of the iterated logarithm; hence $Z_t\to 0$, while $Z_0=1>1\mathrm{/}2$, so the intermediate value theorem gives that the continuous path attains the value $1\mathrm{/}2$ at some finite time and $\tau <\mathrm{\infty }$ almost surely. Brownian law of the iterated logarithm at infinity Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point Brownian motion Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a,b]$ takes every value between $f(a)$ and $f(b)$

$\tau$ is a stopping time. Every path of $Z$ is continuous, so for $t>0$ the event $\{\tau \le t\}$ equals ${\bigcap }_{m\ge 1}{\bigcup }_{q\in \mathbb{Q}\cap [0,t]}\{\mathrm{\mid }{Z}_q-1\mathrm{/}2\mathrm{\mid }<1\mathrm{/}m\}$ identically. A finite infimum of hit times is itself a hit, by continuity and a sequence of hit times decreasing to that infimum. A hit in $[0,t]$ is approximated by rational times; conversely, approximate contacts $q_m\in [0,t]$ have a convergent subsequence by Bolzano–Weierstrass, and continuity gives a hit at its limit. AC permits these countable selections, with positive indices reindexed from zero if required. At $t=0$ the hit event is empty since $Z_0=1$. Every displayed event is ${\mathcal{F}}_t$-measurable, so no null-set transfer is needed. Continuous-time stopping times and stopped sigma-algebras Bolzano-Weierstrass: every bounded real sequence has a convergent subsequence The rationals embed densely in the reals Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point

Finite-grid sampling and maximal domination. The restriction of the all-pairs martingale $Z$ to a finite deterministic grid is a discrete martingale; extend it constantly after the last index. The bounded discrete optional-sampling theorem and the discrete Doob $L^2$ inequality then apply on that grid. Monotone convergence applies to the increasing squares of maxima over nested grids, and Cauchy–Schwarz turns the resulting $L^2$ bound into an integrable dominating supremum. The grid passage is proved in step 1.1. Optional sampling for bounded stopping times Doob Lp maximal inequality Monotone convergence for the integral Cauchy-Schwarz for random variables Dominated convergence Continuous-time filtrations and all-pairs martingales

Uniform integrability and $L^1$ limits. If a sequence converges almost surely and is uniformly integrable, then it converges in $L^1$ (a.s. convergence implies probability convergence by dominated convergence of the error-event indicators). Hence its expectations converge to that of the limit. Uniform integrability plus convergence in probability implies $L^1$ convergence A uniformly integrable family Convergence in probability

AC bookkeeping. AC is inherited from the Brownian and conditional-expectation interfaces and permits the countable selections of hit times and rational approximate contacts in [F3]. The Axiom of Choice