be a finite time horizon. We denote by $\F=(\Fc_t)_{0\leq t\leq T}$ the filtration generated by the random measure $\mu (dt,dz)$.\
By definition of the intensity $\pi(dz)dt$, the compensated jump process: $$\begin{aligned}
\tilde \mu(dt,dz):= \mu(dt,dz)-\pi(dz)dt\end{aligned}$$ is such that $\{\tilde \mu([0,t]\times B),0\leq t\leq T\}$ is a $(P,\F)$ martingale for all $B\in \Cc$, where $\Cc$ is the Borel $\sigma$-field on $C$.\
An insurance strategy is a predictable process $\theta=(\theta_t)_{0\leq t\leq
  T}$ which represents the rate of insurance covered by the agent. We assume that the insurance premium is an affine function of the insurance strategy. Given an initial wealth $x\geq 0$ at time t and an insurance strategy $\theta$, the wealth process of the agent at time $s\in [t,T]$ is then given by : $$\begin{aligned}
\label{eds}
X_s^{t,x,\theta} &:=& x + \int_t^s \left( \alpha - \beta (1-\theta_u)
\right) du - \int_t^s \int_C\theta_u  z \mu(du,dz).\end{aligned}$$ We assume that $\alpha\geq \beta \geq 0$ which means that the premium rate received by the agent is lower then the premium rate paid to the insurer. In the literature, this problem is known as a proportional reinsurance one. The agent is an insurer who has to pay a premium to the reinsurer. We impose that the insurance strategy satisfies: 