\label{valeurm}
\tilde v(t,y) :=\inf_{Y\in\Yc^0(t)} E\left[ \tilde U(yY^{\rho,D}_T) + 
\int_t^T  yY^{\rho,D}_u (\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\,
\pi(dz))_{+})du \right], \end{aligned}$$ We shall adopt a dynamic programming principle approach to study the dual value function . We recall the dynamic programming principle for our stochastic control problem: for any stopping time $0\leq \tau\leq T$, $0\leq t\leq T$ and $0\leq h\leq T-t$, $$\begin{aligned}
\label{pppd}
\tilde v(t,y)&=&\Inf_{Y^{\rho,D}\in \Yc^0(t)}
E\left[\tilde v\left((t+h)\wedge \tau,Y^{\rho,D}_{(t+h)\wedge \tau}\right)\right.\\
&+&\left.\int_{t}^{(t+h)\wedge
    \tau}Y^{\rho,D}_u\left(\alpha-\beta+\left(\beta-\int_C\rho_u(z)\,z\,
      \pi(dz)\right)_+\right)du\right],\nonumber \end{aligned}$$ where $a\wedge b=\min(a,b)$ ( see e.g. Fleming and Soner [@fleson]).\

Let $t\in [0,T]$ and $Y^{\rho,D}\in \Yc^0(t)$. Then the process $Y^{\rho,D}$ evolves according to the following stochastic differential equation $$\begin{aligned}
\label{etat}
dY^{\rho,D}_s&=&Y_{s^-}^{\rho,D}\Big(-dL_s+\int_C(\rho_s(z)-1)\tilde \mu(ds,dz)\Big),\end{aligned}$$ with $$\begin{aligned}
\label{controll}
dL_s=-\frac{dD_s}{D_s}1_{\{D_s>0\}},\,\, t\leq s\leq T,\,\,L_{t^-}=0.\end{aligned}$$

[**Proof.**]{} By Itô’s product rule, we have $$\begin{aligned}
dY^{\rho,D}_s
=Z_{s^-}
dD_s
+D_{s^-}dZ_s
+\triangle Z_s\triangle D_s.\end{aligned}$$ From Equation , we have $$\begin{aligned}
\triangle Z_s\neq 0\,\,\,\mbox{iff} \,\,\, s=T_i=\inf\{u\geq t\,\, \mbox{s.t.} N_u=i\},\,\, i\in \N.\end{aligned}$$ Repeating the same argument as in equation , we have $\triangle Z_s\triangle D_s=0$ $ds\otimes dP \mbox{ a.s. }$ and so equation is proved.\
\
We denote by $\Lc_t$ the set of adapted processes $(L_s)_{t\leq s\leq T}$ with possible jump at time $s=t$ and satisfying equation .\
The Hamilton Jacobi Bellman Variational Inequality arising from the dynamic programming principle is written 