we use the lower semi-continuous envelope of the Hamiltonian H. In fact to prove that the dual value function $\tilde v$ is viscosity solution of our HJBVI, we need only the regularity of $H_*$ to derive inequality .

The following theorem relates the dual value function $\tilde v$ to the HJBVI .

\[viscosite\] The dual value function $\tilde v$ is a viscosity solution of (\[HJB2\]) in $[0,T)\times (0,\infty)$.

[**Proof.**]{} See Appendix.\
\
The HJBVI associated to our problem does not provide a complete characterization of the dual value function $\tilde v$. We need to specify the terminal condition. From the definition of $\tilde v$, it’s obvious that $\tilde v(T,y)=\tilde U(y)$. Since we use the notion of discontinuous viscosity solutions, we need to characterize $ \tilde v^*(T,y)$ and $ \tilde v_*(T,y)$ for all $y\in (0,\infty)$ which is the object of the following theorem.

The terminal conditions of the upper and lower semi-continuous envelope of $\tilde v$ satisfy the following inequalities $$\begin{aligned}
\label{upperterminal}
\tilde v^*(T,y):=\Limsup_{\stackrel{(t^\prime,y^\prime)\rightarrow (T,y)}{t^\prime\in [0,T),\,y^\prime>0}}
\tilde v(t^\prime,y^\prime)\leq \tilde U(y) ,\,\,\mbox{ for all } y\in (0,\infty),\end{aligned}$$ $$\begin{aligned}
\label{lowerterminal}
\tilde v_*(T,y):=\Limsup_{\stackrel{(t^\prime,y^\prime)\rightarrow (T,y)}{t^\prime\in [0,T),\,y^\prime>0}}
\tilde v(t^\prime,y^\prime)\geq \tilde U(y) ,\,\,\mbox{ for all } y\in (0,\infty).\end{aligned}$$

[**Proof.**]{} We first prove inequality . Suppose on the 