as $$\begin{aligned}
\label{HJB2}
&&\min \left \{ 
\frac{\partial \tilde v}{\partial t}(t,y) +
H\left(t,y,\tilde v, \Dy {\tilde v}\right),-\Dy {\tilde v}(t,y)
\right\}=0,\,(t,y)\in [0,T)\times (0,\infty),\end{aligned}$$ with terminal condition $$\begin{aligned}
\label{HJB2t}
\tilde v(T,y)=\tilde U(y)\,,y\in (0,\infty),\end{aligned}$$ where $$H\left(t,y,\tilde v, \Dy {\tilde v}\right):=\Inf_{\rho \in \Sigma}\left\{A^\rho\left(t,y,\tilde v, \Dy {\tilde v}\right)+
y\left(\alpha-\beta+(\beta-\int_C\rho(z)\,z\, \pi(dz))_+\right)\right\},$$ $$\begin{aligned}
A^\rho\left(t,y,\tilde v, \Dy {\tilde v}\right) :=
\int_C\left( \tilde v(t,\rho(z) y)- \tilde v(t,y)-(\rho(z) -1)y \Dy {\tilde v}(t,y)\right)\pi(dz), \end{aligned}$$ and $\Sigma:=\left\{\rho 
\mbox{ positive Borel function defined on }C 
\mbox{ s.t.} 
 \int_C\Big(|\log\rho(z)| + \rho(z) \Big)\pi(dz) <\infty
\right\}$. This divides the time-space solvency region $[0,T)\times (0,\infty)$ into a no-jump region $$\begin{aligned}
R_1=\left \{
  (t,y)\in [0,T]\times (0,\infty),\mbox{ s.t. }\displaystyle{\frac{\partial \tilde v}{\partial t}(t,y) +
H\left(t,y,\tilde v, \Dy {\tilde v}\right)=0}
\right\}\end{aligned}$$ and a jump region $$\begin{aligned}
R_2=\left \{
  (t,y)\in [0,T]\times (0,\infty),\mbox{ s.t. }\displaystyle{\frac{\partial \tilde
      v}{\partial y}(t,y)}=0\right\}.\end{aligned}$$

Viscosity solution
==================

In this section, we provide a rigorous characterization of the dual value function $\tilde v$ as a viscosity solution to the HJBVI (\[HJB2\]). The function $\tilde v$ is not known to be continuous and so we shall work with the notion of discontinuous viscosity solutions. We prove that the dual value function lies in the set of functions $D_{\gamma}([0,T]\times (0,\infty))$ defined as follows: $$\begin{aligned}
D_{\gamma}([0,T]\times (0,\infty)):=\Big\{\displaystyle{f:[0,T]\times (0,\infty)\rightarrow\R}\mbox{ such that },\\
\,\Sup_{y>0} \frac{| f(t,y)|}{y+y^{-\gamma}}<\infty 