Intensity Process for a Pure Jump L\'evy Structural Model with Incomplete Information

In this paper we discuss a credit risk model with a pure jump L\'evy process for the asset value and an unobservable random barrier. The default time is the first time when the asset value falls below the barrier. Using the indistinguishability of the intensity process and the likelihood process, we prove the existence of the intensity process of the default time and find its explicit representation in terms of the distance between the asset value and its running minimal value. We apply the result to find the instantaneous credit spread process and illustrate it with a numerical example.