Hamiltonian $H$ in his arguments is not obvious, we define lower semi-continuous envelope of $H$ by\
$H_*(t,y,\psi,\Dy \psi):=
\Liminf_{\stackrel{(t^\prime,y^\prime)\rightarrow (t,y)}{t^\prime\in [0,T),\,y^\prime>0}}H(t^\prime,y^\prime,\psi,\Dy
\psi)$ for all $(t,y)\in [0,T)\times (0,\infty)$.\
\
Adapting the notion of viscosity solutions introduced by Crandall and Lions [@cralio86] and then by Soner [@son86b]for first integrodifferential operators, we define the viscosity solution as follows:

\(i) A function $\phi$ is a viscosity supersolution of (\[HJB2\]) in $[0,T)\times (0,\infty)$ if $$\begin{aligned}
\label{superdef}
&&\min \left\{
\frac{\partial \psi}{\partial t}(\bar t,\bar y) +
H_*\left(\bar t,\bar y,\psi, \Dy \psi \right), -\Dy \psi(\bar t,\bar y) \right\} \leq 0, \end{aligned}$$ whenever $\psi\in C^1([0,T]\times (0,\infty))$ and $\phi_*- \psi$ has a strict global minimum at $(\bar t,\bar y)\in [0,T)\times 
(0,\infty)$.\
(ii) A function $\phi$ is a viscosity subsolution of (\[HJB2\]) in $[0,T)\times (0,\infty)$ if $$\begin{aligned}
\label{subdef}
&&\min \left\{
\frac{\partial \psi}{\partial t}(\bar t,\bar y) +
H\left(\bar t,\bar y,\psi, \Dy \psi \right), -\Dy \psi(\bar t,\bar y) \right\} \geq 0, \end{aligned}$$ whenever $\psi\in C^1([0,T]\times (0,\infty))$ and $\phi^*-
\psi$ has a strict global maximum at $(\bar t,\bar y)\in [0,T)\times (0,\infty)$.\
(iii) A function $\phi$ is a viscosity solution of (\[HJB2\]) in $[0,T)\times
(0,\infty)$ if it is both super-solution and sub-solution in $[0,T)\times (0,\infty)$.

As it is seen in the definition of viscosity solutions, 