and using the optional decomposition under constraints of Föllmer and Kramkov [@folkab97], Mnif and Pham [@mnipham01] gave the following dual characterization of the set $\Cc_+(t,x)$ $$\begin{aligned}
\label{carac}
& &H\in \Cc_+(t,x)\\ 
&\Longleftrightarrow& 
J(H):=\Sup_{ Z\in \Pc^0(t,x)\,,\tau\in \Tc_t}  
E\left[Z_TH  1_{\tau = T} - \int_t^\tau Z_u (\alpha -\beta +(\beta -\int_C\rho_u(z)\,z\, \pi(dz))_{+})du
\right]
\leq x\nonumber,\end{aligned}$$ where $\Pc^0(t,x)$ is the subset of elements $P^\rho \in \Pc(\Sc(t,x))$ such that $\tilde A_T^{\Sc(t,x)}(P^\rho)$ is bounded and $\Tc_t$ is the set of all stopping times valued in $[0,T]$.\
As a corollary (see their corollary 4.1 ), they deduce that the set of admissible insurance strategies $\Ac(0,x)$ is non empty iff $ x\geq
b:=(\beta-\alpha)T$.

If the agent initial wealth is equal to $ x=b$ and since $\Ac(0,x)$ is not empty, then the wealth process is given by $X_t^{0,x,\theta}=(\beta-\alpha)(T-t)$ for all $0\leq t\leq T$ and so the only admissible strategy is $\theta_t=0$ for all $0\leq t\leq T$ which implies that the dynamic version of the value function satisfies $v\left(t,(\beta-\alpha)(T-t)\right)=0$ for all $0\leq t\leq T$. These boundary conditions obtained from the duality approach are not obvious from the primal approach.

Now, we fix some initial wealth $x\geq b$. We make the following assumption

\[hyputil\] We assume that there exist $\bar \gamma \in (0,1)$, 