I am reading this paper about stochastic differential equations with reflecting boundary conditions. In page 165, an example equation with an explicit solution is presented. However, I can't see that condition 2.3 in the same page, namely:
The set $\\{ t\in \mathbb{R}^+ : \xi(t)\in D\\}$ has $\mathrm{d}|\varphi|$-measure zero.
holds for every instance of such example. Taking $\xi$ to be any continuous path such that $\xi[0,t_0] \subseteq \partial D$, $\xi(t_0, +\infty) \subseteq D$, the total variation $|\varphi|$ would be constant and positive in $[t_0, +\infty)$, so
$$ \int_{t_0}^{+\infty} \mathrm{d}|\varphi|(s) = \int_{t_0}^{+\infty} |\varphi|(s) \mathrm{d}s > 0$$
and this would be a counterexample. What am I reading incorrectly?
