\\
& \mbox{and} & Y_.X_.- \int_t^. Y_u(\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, \pi(dz))_{+})du\end{aligned}$$ are supermartingales under $P$.

[**Proof.**]{} By definition of $\Pc^0(t,x)$, the process $Z_.(X_.- \int_t^.(\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, \pi(dz))_{+})du)$ is a $P$-local supermartingale. From Theorem VII.35 in Dellacherie and Meyer [@delmey82], the process $$\begin{aligned}
 \label{M}
M= Z_.X_.- \int_t^.Z_u(\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, \pi(dz))_{+})du\end{aligned}$$ is a $P$-local supermartingale. Moreover, $M$ is bounded from below by the random variable $-\int_t^.Z_u(\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, \pi(dz))_{+})du)$, which is integrable under $P$. We deduce by Fatou’s lemma that $M$ is a $P$-supermartingale. On the other hand, by Itô’s product rule and since $D$ is predictable with finite variation, we get : $$\begin{aligned}
 \label{surmarD}
& &Y_TX_T- \int_t^T Y_u(\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, \pi(dz))_{+})du\nonumber\\
&=& Y_tX_t + \int_t^T D_{u^-} dM_u+ 
\int_t^T  Z_{u^-} X_{u^-} dD_u + \Sum_{t\leq u\leq T} \triangle M_u \triangle D_u. \end{aligned}$$ From Equation , we have $$\begin{aligned}
\triangle M_s\neq 0 \,\,\,\mbox{iff} \,\,\, s=T_i=\inf\{u\geq t\,\, \mbox{s.t.} N_u=i\},\,\, i\in \N\end{aligned}$$ where $T_i$ is the time of the claim number $i$ and $N$ is the Poisson process representing the number of claims. If the stopping time $T_i$ is accessible, then from Theorem 15.1 in Rogers and Williams [@roge], we have $Z_{T_i^-}=Z_{T_i}$ which is false and so $(T_i)_{i\in \N}$ is a sequence 