not proportional to wealth. An explicit solution to the problem is provided by Bryis by writing the associated Hamilton-Jacobi-Bellman (HJB in short) equation. In Bryis [@bri86] and Gollier [@gol94], they modeled the insurance premium by an affine function of the insurance strategy $\theta=(\theta_t)_{t\in [0,T]}$ which is the rate of insurance decided to be covered by the agent. If the agent is subject to some accident at time $t$ which costs an amount $Z$, then he will pay $\theta_t Z$ and the insurance company reimburses the amount $(1-\theta_t)Z$. They didn’t assume any constraint on the insurance strategy which is not realistic.\
In risk theory, Hipp and Plum [@hipplu00] analysed the trading strategy, in risky assets, which is optimal with respect to the criterion of minimizing the ruin probability. They derived the HJB equation related to this problem and proved the existence of a solution and a verification theorem. When the claims are exponentially distributed, the ruin probability decreases exponentially and the optimal amount invested in risky assets converges to a constant independent of the reserve level. Hipp and Schmidli [@hipplu01] have obtained the asymptotic behaviour of the ruin probability under the optimal investment strategy in the small claim case. Schmidli [@sch99] 