  In this paper, we consider the problem of minimizing the ruin probability of
an insurance company in which the surplus process follows the Sparre Andersen
model. Similar to Bai et al. \cite{bai2017optimal}, we recast this problem in a
Markovian framework by adding another dimension representing the time elapsed
since the last claim. After Markovization, We investigate the regularity
properties of the value function and state the dynamic programming principle.
Furthermore, we show that the value function is the unique constrained
viscosity solution to the associated Hamilton-Jacobi-Bellman equation. It
should be noted that there is no discount factor in our paper, which makes it
tricky to prove the uniqueness. To overcome this difficulty, we construct the
strict viscosity supersolution. Then instead of comparing the usual viscosity
supersolution and subsolution, we compare the supersolution and the strict
subsolution. Eventually we show that all viscosity subsolution is less than the
supersolution.
