the HJBVI contains an inf on a unbounded set which makes it discontinuous. In this paper, we prove the comparison theorem only when the space of claims is a finite one.\
The paper is organized as follows. Section 2 describes the model. In Section 3, we formulate the dual optimization problem and we derive the associated HJBVI for the value function. In Section 4, we prove that the dual value function is a viscosity solution of our HJBVI. In Section 5, we prove a comparison theorem.

Problem formulation
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Let $(\Omega,\Fc,P)$ be a complete probability space. We assume that the claims are generated by a compound Poisson process. More precisely, we consider an integer-valued random measure $\mu (dt,dz)$ with compensator $\pi(dz)dt$. We assume that $\pi(dz)=\varrho G(dz)$ where $G(dz)$ is a probability distribution on the bounded set $C\subseteq \R_+$ and $\varrho$ is a positive constant. In this case, the integral, with respect to the random measure $\mu (dt,dz)$, is simply a compound Poisson process: we have $ 
\int_0^t \int_C z\mu(du,dz)=\sum_{i= 1}^{N_t}Z_i$, where $N=\{N_t,t\geq 0\}$ is a Poisson process with intensity $\varrho$ and $\{Z_i,i\in \N\}$ is a sequence of random variables with common distribution $G$ which represent the claim sizes.\
Let $T>0$ 