with left (resp. right) hand limits in $x$. Moreover, we have the relation $\hat{L}^X = (L_+^X + L_-^X)/2$.\
The local times fulfil $$\label{eqn:int_wrt_loc_time}
    \int_0^t {\text{\large{$\mathds{1}$}}}_{\{y\}}(X_s) \, L^X({\mbox{\upshape d}}s,y) = L^X(t,y)\,, \qquad t < S, \ y \in \mathbb{R}, \ \mathbf{P}\text{-a.s.},$$ $$\label{eqn:loc_time_and_variation_process}
    L_+^X(t,y) - L_-^X(t,y) = 2 \int_0^t {\text{\large{$\mathds{1}$}}}_{\{y\}}(X_s) \, {\mbox{\upshape d}}V_s\,, \qquad t < S, \, y \in \mathbb{R}, \ \mathbf{P}\text{-a.s.},$$ $$\label{eqn:loc_time_zero_outside_compact_interval}
    L^X(t,y) = 0, \qquad t < S, \, y \notin \left[\min_{0 \leq s \leq t} X_s, \max_{0 \leq s \leq t} X_s\right], \ \mathbf{P}\text{-a.s.}$$ and $$\label{eqn:loc_time_as_limit}
    L^X(t,y) = \lim_{\varepsilon \downarrow 0}\frac{1}{\varepsilon} \int_0^t I^y_\varepsilon(X_s) \, {\mbox{\upshape d}}{    \ensuremath{\langle X \rangle}}_s, 
    \qquad t < S, \ y \in \mathbb{R}, \ \mathbf{P}\text{-a.s.},$$ where $$I^y_\varepsilon (x) = {\text{\large{$\mathds{1}$}}}_{[y,y+\varepsilon)}(x) \quad (\text{resp.,} \ {\text{\large{$\mathds{1}$}}}_{(y-\varepsilon, y]}(x), \ 
    \frac{1}{2} \, {\text{\large{$\mathds{1}$}}}_{(y-\varepsilon, y+\varepsilon)}(x)), \qquad x \in \mathbb{R}.$$ In general, SDEs of type (\[eqn:SDE\_mvd\]) admit exploding solutions. Therefore, the convenient state space is the extended real line $\overline{\mathbb{R}} = \mathbb{R}\cup \{-\infty, +\infty\}$ equipped with the $\sigma$-algebra $\mathscr{B}(\overline{\mathbb{R}})$ of Borel subsets. We fix the notion of a solution to Eq. (\[eqn:SDE\_mvd\]) in the following

\[def:solution\] A continuous $(\overline{\mathbb{R}}, \mathscr{B}(\overline{\mathbb{R}}))$-valued stochastic process $(X,\mathbb{F})$ defined on a probability space $(\Omega,\mathcal{F},\mathbf{P})$ is called a solution to Eq. (\[eqn:SDE\_mvd\]) if the following conditions 