Expected exit time from an interval

Given: AC, (H), $a,b>0$, a start $x\in (-a,b)$, the canonical shifted process $B^x$ with its usual augmented natural filtration, and the exit time $\tau$ of $(-a,b)$.

Stopping and normalization. Put $C=(-\mathrm{\infty },-a]\cup [b,\mathrm{\infty })$. Every canonical path is continuous, so for each $t\ge 0$, $\{\tau \le t\}={\bigcap }_{m\ge 1}{\bigcup }_{q\in \mathbb{Q}\cap [0,t]}\{\mathrm{dist}(B_q^x,C)<1\mathrm{/}m\}$. A hit yields arbitrarily close rational times; conversely the continuous distance attains its zero infimum on $[0,t]$. Thus the literal exit is a stopping time. Under $P_x$, $W=B^x-x$ is Brownian with the usual Brownian filtration. Dynkin uses its everywhere-continuous zero-start normalization; it agrees with $W$ on $\{B_0^x=x\}$, a common probability-one event, so all path evaluations and integrals agree there. Continuous-time stopping times and stopped sigma-algebras Brownian motion started at x Natural and usual augmented Brownian filtrations

Dynkin formula. If $f\in C_c^2(\mathbb{R})$ and $\sigma$ is a bounded stopping time, then $E_x[f(B_{\sigma }^x)]=f(x)+E_x{\int }_0^{\sigma }\frac{1}{2}{f}^{\mathrm{\prime }\mathrm{\prime }}(B_s^x)ds$. Dynkin formula for bounded Brownian stopping Brownian motion started at x

Cutoff extension of a quadratic. For $u(y)=(y+a)(b-y)$ there is $f\in C_c^2(\mathbb{R})$ with $f=u$ on a neighbourhood of $[-a,b]$ and $f^{\mathrm{\prime }\mathrm{\prime }}=-2$ there: choose $R>\max (a,b)$ and multiply $u$ by ${\chi }_R$, the smooth compactly supported cutoff equal to $1$ on $[-R,R]$, using Explicit compactly supported smooth cutoffs; the resulting function is $C_c^{\mathrm{\infty }}$, hence $C_c^2$, and therefore bounded. The spaces $C_c({\mathbb{R}}^n)$ and $C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$

Finiteness of the exit time and endpoint values. The path of $B^x$ is continuous, $u(x)>0$ at the start and $u(-a)=u(b)=0$. By Two-sided Brownian exit probability, $B^x$ reaches $b$ before $-a$ with probability $(x+a)\mathrm{/}(a+b)$. The reflected process $-B^x$ is Brownian motion started at $-x$ by Brownian motion, so the same theorem on $(-b,a)$ gives probability $(b-x)\mathrm{/}(a+b)$ that $B^x$ reaches $-a$ before $b$. These disjoint events have probabilities summing to $1$, hence $\tau <\mathrm{\infty }$ almost surely; on $\{\tau <\mathrm{\infty }\}$ continuity gives $B_{\tau }^x\in \{-a,b\}$ and $u(B_{\tau }^x)=0$. Brownian motion started at x 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

Convergence tools. Dominated convergence applies to bounded sequences of random variables; monotone convergence applies to nondecreasing nonnegative sequences, so $E(\tau \wedge n)\uparrow E\tau$ including the value $+\mathrm{\infty }$. Dominated convergence Monotone convergence for the integral 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

AC bookkeeping. Full AC is declared because the cited Dynkin and conditional-expectation interfaces assume it, and it supplies the Countable Choice used by inherited measure-theoretic interfaces. The Brownian coordinate process, shifted law, usual filtration, standing hypothesis (H), and other data of Dynkin's formula are hypotheses recorded in the Statement and [F0]--[F1], not consequences of AC. The cutoff and integer truncations are explicit. The Axiom of Choice