$\bar Q \in\Pc^0(t,x)$ with density $\bar Z_T = \frac {d\bar Q}{dP}$ satisfying $$\begin{aligned}
\label{integrable}
(\bar Z_T)^{-1}\in L^{\bar p}(P)\end{aligned}$$ for some $\bar p > \frac{\bar\gamma}{1-\bar\gamma}$.

Under Assumption \[hyputil\], Existence and uniqueness of problem are proved (see Mnif and Pham [@mnipham01]). We focus now on the study of the dual formulation.\
The two following lemmas allow us to give an expression of the dual value function. We denote by ${\cal D}_t$ the set of nonnegative, nonincreasing predictable and càdlàg processes $D$ $=$ $(D_s)_{t\leq s\leq T}$ with $D_t$ $=$ $1$, $$\begin{aligned}
\Yc^0(t):=\{Y^{\rho,D}=Z^\rho D,\,Z^\rho \in \Pc^0(t,x),\, D\in {\cal D}_t\}\end{aligned}$$ for all $ t\in [0,T]$ and $L_+^0(\Fc_T)$ is the set of nonnegative $\Fc_T$-measurable random variables.

[We omit the dependence of $\Yc^0(t)$ in the initial wealth $x$, since $x$ is fixed in all the paper. ]{}

[The set ${\cal D}_t$ is introduced in the paper of Elkaroui and Jeanblanc [@elkjea98] in the continuous case. In our case, we must enlarge this set to include càdlàg processes and extend some results of Mnif and Pham [@mnipham01].]{}

\[lemdc\] For all $x$ $\geq$ $b$, $X$ $\in$ $\Sc(t,x)$, $Y$ $=$ $ZD$, $Z$ $\in$ $\Pc^0(t,x)$, $D$ $\in$ $\Dc_t$, the processes : $$\begin{aligned}
& & Z_.X_.- \int_t^. Z_u(\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, \pi(dz))_{+})du, 