for all $n\geq n_1$, $I(y_n)\geq (\beta-\alpha)(T-t_n)$ and so $\Max_{x\geq (\beta-\alpha)(T-t_n) }[U(x)-xy_n]=\tilde U(y_n)$. For $n\geq n_0 \vee n_1$, inequality implies $$\begin{aligned}
\tilde v(t_n,y_n)\geq \tilde U(y_n)-\epsilon.\end{aligned}$$ Sending $n$ to infinity and $\epsilon$ to $0$, we prove inequality.

Uniqueness
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We turn now to uniqueness questions. Our next main result is a comparison principle for discontinuous viscosity solutions to the HJBVI . It states that we can compare a viscosity sub-solution and a viscosity super-solution to the HJBVI on $[0,T)\times (0,\infty)$, provided that we can compare them at terminal date as usual in parabolic problems.\
The main difficulty in the comparison theorem comes from the discontinuity of the Hamiltonian. Here, we assume that the claims take values in the set $C=\{\delta_1,\delta_2,...,\delta_d\}$, $\delta_i>0$, $1\leq i\leq d$. In this case the Hamiltonian contains an inf on a bounded set which makes it continuous. We denote by $\pi_i$, $1\leq i\leq d$ the intensity of the Poisson process associated to the claim having the size $\delta_i$. The set $\Sigma$ is defined as follows : $$\begin{aligned}
\label{sigmaprime}
\Sigma:=\left\{ \rho =(\rho_i)_{1\leq i\leq d},\,\, \rho_i>0,\,1\leq i\leq d\right\}.\end{aligned}$$ The Hamiltonian $H$ is given by $$\begin{aligned}
\label{tildeH}
H\left(t,y,\tilde v, \Dy {\tilde v}\right)=\Inf_{\rho \in \Sigma}\left\{A^\rho\left(t,y,\tilde v, \Dy {\tilde v}\right)+
y\left(\alpha-\beta+(\beta-\Sum_{i=1}^d\rho_i \delta_i \pi_i)_+\right)\right\},\end{aligned}$$ where 