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The problem of modeling a default time is well represented in the literature. There are two main approaches: either the default time τ is a stopping time in the asset’s filtration or it is a stopping time in a larger filtration (see Coper and Martin 1996 for a comparison between these approaches). In the first approach, the so-called structural form pioneered by Merton (1974), the default time τ is a stopping time in the filtration of the prices. In the second case, the idea is also to compute the value of the defaultable claim, and it is fruitful to introduce the notion of intensity of the default. The intensity of the default time acts as a change of the spot interest rate in the pricing formula. In this work, we consider the so-called \-model. It is a one-default model which gives the conditional law of a random time with respect to a reference filtration. In this paper, we work on a stochastic differential equation (called equation (\) below); this equation plays an essential role in this article, but its application has been submitted to a hypothesis of continuity. Then it is important to know under what conditions the hypothesis of continuity is satisfied. This is the main motivation of our research.

The Application of Kolmogorov’s theorem in the one-default model