A DIRECT APPROACH TO A FIRST-PASSAGE PROBLEM WITH APPLICATIONS IN RISK THEORY

In this article, we consider a risk process which exbihits the key features of companies with steady outflows and sporadic inflows (e. g., discoveries, patents). A risk management policy is further implemented stating that the outflow rate is reduced when no revenue (inflow) is generated within an Erlang-n time period. For the surplus process of interest, a Markovian representation is first given which leads to the form of the solution for the Laplace transform of the time to ruin. A homogeneous linear integro-differential equation for the Laplace transform of the time of ruin is later derived. The boundary conditions of the aforementioned integro-differential equation are used to complete the representation of the Laplace transform of the time to ruin. Finally, numerical applications are considered to illustrate the effectiveness of this risk management policy to lower the company's solvency risk.