Minimizing ruin probability under the Sparre Anderson model

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. We recast this problem in a Markovian framework by adding another variable 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. Since the discount factor is not included in this model, the proof of uniqueness of the viscosity solution is tricky. To overcome this difficulty, we construct the strict viscosity supersolution. Instead of comparing the usual viscosity supersolution and subsolution, we compare the strict supersolution and the subsolution. Eventually, we show that any viscosity subsolution is less than the supersolution.