are fulfilled:

\(i) $X_0$ is real-valued.

\(ii) $X_t = X_{t \wedge S_\infty^X}$, $t \geq 0$, $\mathbf{P}$-a.s., where $S_\infty^X := \inf\{t \geq 0:|X_t| = +\infty\}$.

\(iii) $(X,\mathbb{F})$ is a semimartingale up to $S_\infty^X$.

\(v) There exists a Wiener process $(B,\mathbb{F})$ such that Eq. (\[eqn:SDE\_mvd\]) is satisfied for all $t < S_\infty^X$ $\mathbf{P}$-a.s.

The Results
===========

We recall that in Eq. (\[eqn:SDE\_mvd\]) $L^X$ stands either for the right, the left or the symmetric local time and we treat these three cases simultaneously in the sequel. For the sake of brevity, in the following we write for example $\{\nu \geq 1/2\}$ instead of $\{y \in \mathbb{R}: \nu(\{y\}) \geq 1/2\}$.

\[lemma:loc\_time\_zero\] Let $(X,\mathbb{F})$ be a solution of Eq. (\[eqn:SDE\_mvd\]) with right (resp., left, symmetric) local time. Then it holds $$\begin{split}
                                                     L_-^X(t,x) = 0, \qquad & t < S_\infty^X, \ x \in \{\nu \geq 1/2\}, \ \mathbf{P}\text{-a.s.} \phantom{\Bigr)}\\
                      \Bigl(\text{resp., } L_-^X(t,x) = 0, \qquad & t < S_\infty^X, \ x \in \{\nu < -1/2\},  \ \mathbf{P}\text{-a.s.}, \\
                                                           L_-^X(t,x) = 0, \qquad & t < S_\infty^X, \ x \in\{|\nu| > 1 \text{ or } \nu = 1\}, \ \mathbf{P}\text{-a.s.}\Bigr)
        \end{split}$$ and $$\begin{split}
                                                     L_+^X(t,x) = 0, \qquad & t < S_\infty^X, \ x \in \{\nu > 1/2\}, 