n\in \mathbb{N}\right\} &
\end{array}
\label{Def._L(U)}$$be the class of predictable processes $\theta \in \mathcal{P}$ integrable with respect to $U^{c}$ in the sense of local martingale, and by $$\Lambda \left( U^{c}\right) :=\left\{ \int \theta _{0}dU^{c}:\theta _{0}\in
\mathcal{L}\left( U^{c}\right) \right\}$$the linear space of processes which admits a representation as the stochastic integral with respect to $U^{c}$. For an integer valued random measure $\mu ^{\prime }$ we denote by $\mathcal{G}\left( \mu ^{\prime
}\right) $ the class of $\widetilde{\mathcal{P}}$-measurable processes $\theta _{1}:$ $\Omega \times \mathbb{R}_{+}\times \mathbb{R}_{0}\rightarrow
\mathbb{R}$ satisfying the following conditions: $$\begin{array}{cl}
\left( i\right) & \theta _{1}\in \widetilde{\mathcal{P}}, \\
\left( ii\right) & \int\limits_{\mathbb{R}_{0}}\left\vert \theta _{1}\left(
t,x\right) \right\vert \left( \mu ^{\prime }\right) ^{\mathcal{P}}\left(
\left\{ t\right\} ,dx\right) <\infty \ \forall t>0, \\
\left( iii\right) & \text{The process } \\
& \left\{ \sqrt{\sum\limits_{s\leq t}\left\{ \int\limits_{\mathbb{R}_{0}}\theta _{1}\left( s,x\right) \mu ^{\prime }\left( \left\{ s\right\}
,dx\right) -\int\limits_{\mathbb{R}_{0}}\theta _{1}\left( s,x\right) \left(
\mu ^{\prime }\right) ^{\mathcal{P}}\left( \left\{ s\right\} ,dx\right)
\right\} ^{2}}\right\} _{t\in \mathbb{R}_{+}}\in \mathcal{A}_{loc}^{+}.\end{array}$$The set $\mathcal{G}\left( \mu ^{\prime }\right) $ represents the domain of the functional $\theta _{1}\rightarrow \int \theta _{1}d\left( \mu ^{\prime
}-\left( \mu ^{\prime }\right) ^{\mathcal{P}}\right) ,$ which assign to $\theta _{1}$ the unique purely discontinuous local martingale $M$ with $$\bigtriangleup M_{t}=\int\limits_{\mathbb{R}_{0}}\theta _{1}\left(
t,x\right) \mu ^{\prime }\left( \left\{ t\right\} ,dx\right) -\int\limits_{\mathbb{R}_{0}}\theta 