Wang and Yan [@HeWanYan] or Theorem III.5.19 in Jacod and Shiryaev .

Denote by $D_{t}:=\mathbb{E}\left[ \left. \frac{d\mathbb{Q}}{d\mathbb{P}}\right\vert \mathcal{F}_{t}\right] $ the càdlàg version of the density process. For the increasing sequence of stopping times $\tau
_{n}:=\inf \left\{ t\geq 0:D_{t}<\frac{1}{n}\right\} $ $n\geq 1$ and $\tau
_{0}:=\sup_{n}\tau _{n}$ we have $D_{t}\left( \omega \right) =0$ $\forall
t\geq \tau _{0}\left( \omega \right) $ and $D_{t}\left( \omega \right) >0$ $\forall t<\tau _{0}\left( \omega \right) ,$ i.e.$$D=D\mathbf{1}_{[\hspace{-0.05cm}[0,\tau _{0}[\hspace{-0.04cm}[},
\label{D=D1[[0,To[[}$$and the process $$\frac{1}{D_{s-}}\mathbf{1}_{[\hspace{-0.05cm}[D_{-}\not=0]\hspace{-0.04cm}]}\text{ is integrable w.r.t. }D,  \label{1/D_integrable_wrt_D}$$where we abuse of the notation by setting $[\hspace{-0.05cm}[D_{-}\not=0]\hspace{-0.04cm}]:=\left\{ \left( \omega ,t\right) \in \Omega \times \mathbb{R}_{+}:D_{t-}\left( \omega \right) \neq 0\right\} .$ Both conditions $\left( \ref{D=D1[[0,To[[}\right) $ and $\left( \ref{1/D_integrable_wrt_D}\right) $ are necessary and sufficient in order that a semimartingale to be an *exponential semimartigale* [@HeWanYan Thm. 9.41], i.e. $D=\mathcal{E}\left( Z\right) $ the Doléans-Dade exponential of another semimartingale $Z$. In that case we have $$\tau _{0}=\inf \left\{ t>0:D_{t-}=0\text{ or }D_{t}=0\right\} =\inf \left\{
t>0:\triangle Z_{t}=-1\right\}.  \label{Tau0=JumpZ=-1}$$

It is well known that the Lévy-processes satisfy the weak property of predictable representation [@HeWanYan], when the completed natural filtration is considered. In the following lemma we present the characterization of the density processes for the case of these processes.

\[Q&lt;&lt;P =&gt;\] Given an absolutely continuous probability 