This is a problem and solution from Rene Schilling's Brownian Motion.
Let $B_t = (b_t,\beta_t), t \ge 0$ be a two dimensional Brownian motion and set $r_t := |B_t| = \sqrt{b_t^2 + \beta_t^2}.$ Show that the stochastic integrals $\int_0^t b_s / r_s db_s$ and $\int_0^t \beta_s / r_s d\beta_s$ exist.
The solution is quite simple and easy. However, I have one doubt. In order for the integrands to be well-defined, we require $r_s >0$. But how do we know this?
Suppose $B$ starts at a point of the plane with norm $1$ (say). For a large $N$ let $\tau_N$ be the first time $r_t$ equals $N$. Then the process $Z_t:=\log N -\log r_{t\wedge\tau_N}$ is a local martingale (Ito's formula), and is bounded below by $0$. It follows that $Z$ is a non-negative supermartingale, and in particular $\log N = \Bbb E[Z_0]\ge \Bbb E[Z_{\tau_0\wedge\tau_N}; \tau_0<\infty]$, where $\tau_0$ is the first time $r_t$ vanishes (if any—$\tau_0$ is taken to be $\infty$ if $r_t$ never gets to $0$.) Because $Z_{\tau_0}=+\infty$ on $\{\tau_0<\tau_N\}$, you must have $\Bbb P[\tau_0<\tau_N]=0$. Now let $N\to\infty$. In brief, the magnitude of two-dimensional Brownian motion is strictly positive for all times, almost surely.