\pi(dz))_{+})du \right].\end{aligned}$$ Since $D^n_u$ $\rightarrow$ $1_{u\leq \tau}$ a.s., for all $t\leq u\leq T$, we have by Fatou’s lemma : $$\begin{aligned}
& & E\left[Z_TH  1_{\tau = T} - \int_0^\tau Z_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_t^T 
Z_uD_u (\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, \pi(dz))_{+})du \right].\end{aligned}$$ Identifying a probability measure $Q$ $\in$ $\Pc^0(t,x)$ with its density process $Z$, we then obtain from Bayes formula : $$\begin{aligned}
J(H) &= & \sup_{Q\in\Pc^0(t,x),\tau\in\Tc_t} 
E^Q\left[H  1_{\tau = T} - \int_0^\tau (\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\, \pi(dz))_{+})du \right]. \end{aligned}$$ Conversely, by the supermartingale property of $Y_.X_.- \int_0^. Y_u(\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, \pi(dz))_{+})du$ for any $Y$ $\in$ $\Yc^0(t)$, see Lemma \[lemdc\], we have : $$\begin{aligned}
 \label{interH}
\sup_{Y\in\Yc^0(t)} 
E \left[Z_T D_T   H  - \int_t^T 
Z_uD_u (\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, \pi(dz))_{+})du \right] \leq x, \; \forall H \in \Cc_+(x).\end{aligned}$$ Now, by characterization , any $H$ $\in$ $L_+^0(\Fc_T)$ lies in $\Cc_+(J(H))$. We then deduce from inequality that $$\begin{aligned}
\sup_{Y\in\Yc^0(t)} 
E \left[Y_T   H  - \int_0^T 
Y_u (\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, \pi(dz))_{+})du \right] &\leq& J(H),\end{aligned}$$ which proves the required equality .\
The dual problem of is written as: $$\begin{aligned}
