${\cal U}_t$ $=$ $\{(\rho_s(z))_{t\leq s\leq T}$ predictable process : $\rho_s(z)$ $>$ $0$, a.s., $t\leq s \leq T$,$z\in C$, $\int_t^T \int_C\Big(|\log\rho_s(z)| + \rho_s(z) \pi(dz)\Big) ds <\infty$ and $E[Z_T^\rho] = 1\}$.\
By Girsanov’s theorem, the predictable compensator of an element $X^\theta \in \Sc(t,x)$ under $P^\rho$ $=$ $Z_T^\rho.P$ is : $$\begin{aligned}
A_s^{\rho,\theta} &=&  \int_t^s (\alpha -\beta)du + \int_t^s \theta_u (\beta-\int_C\rho_u(z)\, z\,\pi(dz)) du.\end{aligned}$$ We deduce from Lemma 2.1 of Föllmer and Kramkov [@folkab97] that $\Pc(\Sc(t,x))$ $=$ $\{P^\rho~: \rho \in {\cal U}_t\}$ and the upper variation process of $P^\rho$ is : $$\begin{aligned}
\tilde A^{\Sc(t,x)}_s(P^\rho) &=& \int_t^s (\alpha -\beta)du +\int_t^s  (\beta-\int_C\rho_u(z)\,z\,\pi(dz))_+ du.\end{aligned}$$ From the non-decreasing property of $U$, we have $$\begin{aligned}
v(t,x)=\Sup_{H\in\Cc_+(t,x)}E[U(H)],\end{aligned}$$ where $\Cc_+(t,x)=\{H\in L^0_+(\Fc_T): X_T^{t,x,\theta} \geq H \,a.s. \mbox{ for
  } 
\theta\in \Ac(t,x)\}$. It is easy to check the convexity property of the family $\Cc_+(t,x)$. For the closure property of this family in the semi-martingale topology, we refer to Pham [@pham00].\
The semi-martingale topology is associated to the Emery distance between two semi-martingales $\tilde X^1$ and $\tilde X^2$ defined as : $$\begin{aligned}
 \label{emery}
D_E(\tilde X^1,\tilde X^2) &:=& \sum_{n\geq 1} 2^{-n} 
E\left[\sup_{0\leq t\leq T\wedge n}|\tilde X_t^1-\tilde X_t^2|\wedge 1 \right].\end{aligned}$$ We refer to Mémin [@m80] for details on the semi-martingale topology. Since $\Cc_+(t,x)$ is convex and closed 