Let \(Z\) be an \(\mathbb{R}\)-valued semi-martingale such that \(Z\) and \(Z_{-}\) do not vanish. Prove that

\[ \mathcal{L}(Z)_{t}=\ln \left(\left|\frac{Z_{t}}{Z_{0}}\right|\right)+\int_{0}^{t} \frac{1}{2 Z_{s-}^{2}} d\left\langle Z^{c}\right\rangle_{s}-\sum_{s \leq t}\left(\ln \left|\frac{Z_{s}}{Z_{s-}}\right|+1-\frac{Z_{s}}{Z_{s-}}\right) . \]