the nonlinear Bellman equation. In few cases they constructed the optimal control. Extensions to the case of constrained investment are considered by Cvitanić and Karatzas [@cvikar92] and to the case of incomplete markets by Karatzas et al. [@karlehshrxu91]. Typically, in incomplete markets, we have to solve a dual problem which leads in the Markov case to nonlinear partial differential equation.\
In this paper, we model the claims by using a compound Poisson process. The insurance trading strategy is constrained to remain in $[0,1]$. We impose a constraint of non-bankruptcy on the wealth process $X_t$ of the agent for all $t$. The objective of the agent is to maximize the expected utility of the terminal wealth over all admissible strategies and to determine the optimal policy of insurance.\
In our case the wealth process positivity constraint is a real one unlike the problem formulated in Touzi [@tou00].\
Our stochastic optimization problem is a particular case of a general structure of problems developed in Mnif and Pham [@mnipham01] who considered the following optimization problem: $$\begin{aligned}
 \label{optimdyn}
\Max_{X\in\Xc_+(x)} E[U(X_T)], \;\;\; x\in \R, \end{aligned}$$ where $\Xc_+(x):=\left\{ x+ X~: X \in \Xc \mbox{ s.t }\,\, X_t\geq 0 \mbox{
    for all } 0\leq t\leq T\right\}$ where 