Theorem 5.2.4: Reuter, {{\cite[Theorem 39.66]{Rogers}}} | Brownian Motion and Stochastic Calculus

Theorem 5.2.4 (Reuter, [Theorem 39.66, RW89]).label Let $(\Omega, \cf, \bp)$ be a filtered probability space, $X: \Omega \to C([0, \infty); \real^{d})$ be a standard $\bracs{\mathcal{F}_t}$-Brownian motion with drift $\mu$ starting at $0$. If $\tau = \inf\bracs{t \ge 0: |X_t| = 1}$, then $X_{\tau}$ and $\tau$ are independent.

Proof. If $\mu = 0$, then $X_{\tau}$ and $\tau$ are independent by rotational invariance.

Otherwise, let $\wien$ be the classical Wiener measure on $C([0, \infty); \real^{d})$, $\mathcal{V}$ be the distribution of $X$, and $Y$ be the canonical process on $C([0, \infty); \real^{d})$. For each $t \ge 0$, let $\mathcal{G}_{t} = \sigma(\bracs{Y_s|0 \le s \le t})$, then by the Cameron-Martin formula,

\[h(t, Y_{t}) = \frac{d \mathcal{V}}{d \wien}\bigg |_{\mathcal{G}_t}= \exp\braks{\angles{\mu, Y_t}_{\real^d} - \frac{1}{2}\norm{\mu}_{\real^d}^2 t}\]

Since $h(t \wedge \tau, X_{t \wedge \tau})$ is a uniformly integrable martingale,

\[\frac{d \mathcal{V}}{d\wien}\bigg |_{\mathcal{G}_{T^+}}= h(\tau, Y_{\tau})\]

In which case, for any measurable functions $f: \mathbb{S}^{d}\to [0, 1]$ and $g: [0, \infty) \to [0, 1]$,

\begin{align*}\ev^{\mathcal{V}}[f(Y_{\tau})g(\tau)]&= \ev^{\wien}\braks{f(Y_\tau)g(\tau)\exp(\angles{\mu, Y_\tau}_{\real^d} - \frac{1}{2}\norm{\mu}_{\real^d}^2 \tau)}\\&= \ev^{\wien}[f(Y_{\tau})e^{\angles{\mu, Y_\tau}_{\real^d}}]\ev^{\wien}[g(\tau)e^{- \frac{1}{2}\norm{\mu}_{\real^d}^2 \tau}] \\&= \ev^{\mathcal{V}}(f(Y_{\tau}))\ev^{\mathcal{V}}(g(\tau))\end{align*}

$\square$