of totally inaccessible stopping times. On the other hand, $D$ is predictable process and so from Lemma 27.3 in Rogers and Williams [@roge] we have $\triangle D_\tau =0$ for every totally inaccessible stopping time $\tau$. Since $\triangle M_s=0$ if $s\neq T_i$ and $\triangle D_s=0$ if $s$ is a totally inaccessible stopping time, we have $$\begin{aligned}
\label{var}
\triangle M_s\triangle D_s=0\,\,ds\otimes dP \mbox{ a.s. }\end{aligned}$$ Since $D$ is nonnegative and nonincreasing, and $Z$, $X$ are nonnegative, this shows that the process : $Y_.X_.- \int_t^. Y_u(\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, \pi(dz))_{+})du$ is a $P$-local supermartingale, bounded from below by an $L^1(P)$ random variable, and hence a $P$-supermartingale.

\[lemelk\] For all $H\in L_+^0(\Fc_T)$, we have : $$\begin{aligned}
 \label{vhegal}
J(H) =
\sup_{Y\in\Yc^0(t)} E\left[Y_TH-
\int_t^T Y_u (\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, \pi(dz))_{+})du\right]\leq x\end{aligned}$$

[**Proof.**]{} Fix some $H$ $\in$ $L^0_+(\Fc_T)$. Given an arbitrary $\tau$ $\in$ $\Tc_t$, we define a sequence $(D^n)_{n}$ of elements in $\Dc_t$ by : $$\begin{aligned}
D^n_s\ &=& \exp\left(-\int^{s}_{t}n1_{\tau \leq u}du\right),\;\;
t\leq s\leq T, \; n \in \N.\end{aligned}$$ We then have for all $Z$ $\in$ $\Pc^0(t,x)$ : $$\begin{aligned}
& &E \left[Z_T D^n_T   H  - \int_t^T 
Z_uD^n_u (\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, \pi(dz))_{+})du \right]\\
& &\leq \sup_{Z\in \Pc^0(t,x),D\in\Dc_t} 
E \left[Z_T D_T   H  - \int_0^T 
Z_uD_u (\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, 