_{1}\left( t,x\right) \left( \mu ^{\prime }\right) ^{\mathcal{P}}\left( \left\{ t\right\} ,dx\right) .$$

We use the notation $\int
\theta _{1}d\left( \mu ^{\prime }-\left( \mu ^{\prime }\right) ^{\mathcal{P}}\right) $ to write the value of this functional in $\theta _{1}$. It is important to point out that this functional is not, in general, the integral with respect to the difference of two measures. For a detailed exposition on these topics see He, Wang and Yan [@HeWanYan] or Jacod and Shiryaev [Jcd&Shry 2003]{}, which are our basic references.

In particular, for the Lévy process $L$ with jump measure $\mu $, $$\mathcal{G}\left( \mu \right) \equiv \left\{ \theta _{1}\in \widetilde{\mathcal{P}}:\left\{ \sqrt{\sum\limits_{s\leq t}\left\{ \theta _{1}\left(
s,\triangle L_{s}\right) \right\} ^{2}\mathbf{1}_{\mathbb{R}_{0}}\left(
\triangle L_{s}\right) }\right\} _{t\in \mathbb{R}_{+}}\in \mathcal{A}_{loc}^{+}\right\} ,  \label{G(miu) Definition}$$since $\mu ^{\mathcal{P}}\left( \left\{ t\right\} \times A\right) =0$, for any Borel set $A$ of $\mathbb{R}_{0}$.

We say that the semimartingale $U$ has the *weak property of predictable representation* when $$\mathcal{M}_{loc,0}=\Lambda \left( U^{c}\right) +\left\{ \int \theta
_{1}d\left( \mu _{U}-\mu _{U}^{\mathcal{P}}\right) :\theta _{1}\in \mathcal{G}\left( \mu _{U}\right) \right\} ,\   \label{Def_weak_predictable_repres.}$$where the previous sum is the linear sum of the vector spaces, and $\mathcal{M}_{loc,0}$ is the linear space of local martingales starting at zero.

Let $\mathcal{M}$ and $\mathcal{M}_{\infty }$ denote the class of càdlàg and 