= Solution
Let $f(y)=|y|$ away from zero. Its first and second <partial derivatives> are
$$
\partial_kf(y)=\frac{y_k}{|y|},\qquad
\partial_{jk}f(y)=\frac{\delta_{jk}}{|y|}-\frac{y_jy_k}{|y|^3}.
$$
Thus its <Laplace operator> is $\Delta f(y)=(n-1)/|y|$. The <quadratic covariations> of the <independent> Brownian coordinates satisfy $d[B_j,B_k]_s=\delta_{jk}ds$. The stopped <Itô formula> therefore gives
$$
\boxed{X_{t\wedge\tau}=|x|+\frac{n-1}{2}\int_0^{t\wedge\tau}\frac{ds}{X_s}
+\sum_{k=1}^n\int_0^{t\wedge\tau}\frac{B_k(s)}{X_s}\,dB_k(s).}
$$
For $t<\tau$, this is the required integral representation.
To identify the driving <Brownian motion>, the stopped stochastic term alone is insufficient: its <quadratic variation> is $t\wedge\tau$. Extend its unit direction after exit. Define a vector <predictable process> $u(s)$ by
$$
u(s)=\begin{cases}B(s)/X_s,&s\leq\tau,\\(1,0,\ldots,0),&s>\tau,\end{cases}
\qquad
W_t=\sum_{k=1}^n\int_0^tu_k(s)\,dB_k(s).
$$
Predictability follows from the continuous adapted direction before stopping and the left-continuous adapted indicator $\mathbf1_{\{s\leq\tau\}}$. This $W$ is a zero-starting <continuous martingale>, and
$$
[W]_t=\int_0^t\sum_{k=1}^nu_k(s)^2ds=t.
$$
The <Lévy characterization of Brownian motion> identifies $W$ as a one-dimensional <Brownian motion>. Before $\tau$, the preceding representation becomes the <stochastic differential equation> of the dimension-$n$ <Bessel process>:
$$
\boxed{dX_t=\frac{n-1}{2X_t}\,dt+dW_t\qquad(t<\tau).}
$$
The fully stopped equation has both integrals stopped at $\tau$.
For $n=2$, apply the one-dimensional <Itô formula> to the <logarithm>. Before $\tau$, $d[X]_t=dt$, so the two drift terms cancel:
$$
d\log X_t=\frac1{X_t}\left(\frac1{2X_t}\,dt+dW_t\right)
-\frac1{2X_t^2}\,dt=\frac1{X_t}\,dW_t.
$$
Consequently,
$$
\boxed{\log X_{t\wedge\tau}=\log|x|+\int_0^{t\wedge\tau}\frac{dW_s}{X_s}
\quad\text{is a bounded martingale}.}
$$
Indeed $X_s\geq r$ before stopping, so the <Itô isometry> bounds the stochastic term's second <moment> by $t/r^2$, and $\log X_{t\wedge\tau}\in[\log r,\log R]$. This is the <martingale> assertion under the paper's standing stopping assumption. Without stopping, the same calculation establishes only a <local martingale>; the stopped qualification is essential for the stated argument.
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