measure $\mathbb{Q}\ll \mathbb{P}$, there exist coefficients $\theta _{0}\in \mathcal{L}\left(
W\right) \ $and $\theta _{1}\in \mathcal{G}\left( \mu \right) $ such that $$\frac{d\mathbb{Q}_{t}}{d\mathbb{P}_{t}}=\frac{d\mathbb{Q}_{t}}{d\mathbb{P}_{t}}\mathbf{1}_{[\hspace{-0.05cm}[0,\tau _{0}[\hspace{-0.04cm}[}=\mathcal{E}\left( Z^{\theta }\right) \left( t\right) ,  \label{Dt=exp(Zt)}$$where $Z_{t}^{\theta }\in \mathcal{M}_{loc}$ is the local martingale given by$$Z_{t}^{\theta }:=\int\limits_{]0,t]}\theta _{0}dW+\int\limits_{]0,t]\times
\mathbb{R}_{0}}\theta _{1}\left( s,x\right) \left( \mu \left( ds,dx\right)
-ds\ \nu \left( dx\right) \right) ,  \label{Def._Ztheta(t)}$$and $\mathcal{E}$ represents the Doleans-Dade exponential of a semimartingale. The coefficients $\theta _{0}$ and $\theta _{1}$ are $dt$-a.s and $\mu _{\mathbb{P}}^{\mathcal{P}}\left( ds,dx\right) $-a.s. unique on $[\hspace{-0.05cm}[0,\tau _{0}]\hspace{-0.04cm}]$ and $[\hspace{-0.05cm}[0,\tau _{0}]\hspace{-0.04cm}]\times \mathbb{R}_{0}$ respectively for $\mathbb{P}$-almost all $\omega $. Furthermore, the coefficients can be choosen with $\theta _{0}=0$ on $]\hspace{-0.05cm}]\tau _{0},\infty \lbrack
\hspace{-0.04cm}[$ and $\theta _{1}=0$ on $]\hspace{-0.05cm}]\tau
_{0},\infty \lbrack \hspace{-0.04cm}[\times \mathbb{R}$ .

*Proof.* We only address the uniqueness of the coefficients $\theta _{0}$ and $\theta _{1},$ because the representation follows from $\left( \ref{D=D1[[0,To[[}\right) $ and $\left( \ref{1/D_integrable_wrt_D}\right) .$ Let assume, that we have two possible vectors $\theta :=\left(
\theta _{0},\theta _{1}\right) $ and $\theta ^{\prime }:=\left( \theta
_{0}^{\prime },\theta _{1}^{\prime }\right) $ satisfying the representation, i.e. $$\begin{array}{rl}
D_{u}\mathbf{1}_{[\hspace{-0.05cm}[0,\tau _{0}[\hspace{-0.04cm}[} & =\int
D_{t-}d\{\int\limits_{]0,t]}\theta _{0}\left( s\right)
dW_{s}+\int\limits_{]0,t]\times \mathbb{R}_{0}}\theta _{1}\left( s,x\right)
\left( \mu \left( ds,dx\right) -ds\ \nu \left( dx\right) \right) \} \\ 
& =\int D_{t-}d\{\int\limits_{]0,t]}\theta _{0}^{\prime }\left( s\right)
dW_{s}+\int\limits_{]0,t]\times \mathbb{R}_{0}}\theta _{1}^{\prime }\left(
