studied the optimal proportional reinsurance policy which minimizes the ruin probability in infinite horizon. He derived the associated HJB equation, proved the existence of a solution and a verification theorem in the diffusion case. He proved that the ruin probability decreases exponentially whereas the optimal proportion to insure is constant. Moreover, he gave some conjecture in the Cramér-Lundberg case. Højgaard and Taksar [@hoj98] studied another problem of proportional reinsurance. They considered the issue of reinsurance optimal fraction, that maximizes the return function. They modelled the reserve process as a diffusion process.\
Touzi [@tou00] studied the problem of maximizing the expected utility from terminal wealth when the insurance strategy is valued in $[0,1]$ at each time . He modeled the wealth process by a Doléans-Dade exponential process. He assumed a boundedness assumption on the jump term which guarantees the positivity of the wealth process. He solved this stochastic control problem by using duality method.\
Duality method was introduced by Karatzas et al. [@karle] and Cox and Huang [@cox89]. Cox and Huang characterized the optimal consumption- portfolio policies when there exist non-negativity constraints on consumption and on final wealth. They gave a verification theorem which involves a linear partial differential equation unlike 