problem
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First we introduce some notations. Let $x\geq 0$ and $t\in[0,T]$. We denote by $\Pc(\Sc(t,x))$ the set of all probability measures $Q$ $\sim$ $P$ with the following property: there exists $A$ $\in$ $\Ic_p$, set of non-decreasing predictable processes with $A_0$ $=$ $0$, such that : $$\begin{aligned}
 \label{pc}  
X - A  \; \mbox{is a} \; Q-\mbox{local super-martingale for any} \; 
X \in \Sc(t,x).  \end{aligned}$$ The upper variation process of $\Sc(t,x)$ under $Q$ $\in$ $\Pc(\Sc(t,x))$ is the element $\tilde{A}^{\Sc(t,x)}(Q)$ in $\Ic_p$ satisfying and such that $A-{\tilde A^{\Sc(t,x)}}(Q)$ $\in$ $\Ic_p$ for any $A$ $\in$ $\Ic_p$ satisfying .\
From Lemma 2.1 of Föllmer and Kramkov [@folkab97], we can derive $\Pc(\Sc(t,x))$ and ${\tilde A^{\Sc(t,x)}}(Q)$. This result states that $Q\in\Pc(\Sc(t,x))$ iff there is an upper bound for all the predictable processes arising in the Doob-Meyer decomposition of the special semi-martingale $V$ $\in$ $\Sc(t,x)$ under $Q$. In this case, the upper variation process is equal to this upper bound.\
It is well-known from the martingale representation theorem for random measures (see e.g. Brémaud [@bre81]) that all probability measures $Q$ $\sim$ $P$ have a density process in the form : $$\begin{aligned}
\label{exporetiel} 
Z^\rho_s &=& \Ec \left( \int_t^s\int_C (\rho_u(z) -1) \tilde \mu(du,dz) \right),\,\, s\in [t,T],\end{aligned}$$ where $\rho$ $\in$ 