is not difficult to check that $L$ and $R$ are just the symmetric local times of $X$ in $r_1$ and $r_2$, respectively. Our results, especially Corollary \[cor:reflection\_and\_absorbing\](i), now show that Eq. (\[eqn:SDE\_refl\_schmidt\]) even coincides with Eq. (\[eqn:SDE\_mvd\]) for the diffusion coefficient $b$, symmetric local time, drift measure $\nu = \delta_{r_1} - \delta_{r_2}$ and initial condition $X_0 \in [r_1,r_2]$, i.e., with the equation $$\label{eqn:sde_refl}
        X_t = X_0 + \int_0^t b(X_s) \, {\mbox{\upshape d}}B_s + \hat{L}^X(t,r_1) - \hat{L}^X(t,r_2), \qquad X_0 \in [r_1,r_2]\,.$$ For Eq. (\[eqn:sde\_refl\]) the condition (\[eqn:SDE\_refl\_schmidt\])(ii) needs not be specified since it is satisfied for any solution which follows via Corollary \[cor:reflection\_and\_absorbing\](i). Moreover, (\[eqn:SDE\_refl\_schmidt\])(iii) holds because of (\[eqn:int\_wrt\_loc\_time\]).

\(iii) Similar as in the preceding remark the non-negativity condition $X_t \geq 0$, $t \geq 0$, of the solution to the equation $$X_t = X_0 + \int_0^t {\text{\large{$\mathds{1}$}}}_{(0,+\infty)}(X_s) \, {\mbox{\upshape d}}B_s + a \, \int_0^t {\text{\large{$\mathds{1}$}}}_{\{0\}} (X_s) \, {\mbox{\upshape d}}s,$$ with $X_0 \geq 0$ and $a \geq 0$, which is studied in R. Chitashvili [@chitashvili], can be dropped. More detailed, for a solution $(X,\mathbb{F})$ of this equation by (\[eqn:loc\_time\_as\_limit\]) it follows $L_-^X(t,0) = 0$, $t \geq 0$, $\mathbf{P}$-a.s. Combined with (\[eqn:loc\_time\_and\_variation\_process\]) this implies $$\frac{1}{2}\, L_+^X(t,0) = a \, \int_0^t {\text{\large{$\mathds{1}$}}}_{\{0\}} (X_s) \, 