6 is devoted to the proof of the Markov property of the information process with respect to the filtration $\mathbb{F}^{\beta}$, the smallest filtration which contains $\mathbb{F}^0$ and satisfies the usual conditions of right-continuity and completeness. In Section 7 we show that the process $\beta$ is a semimartingale and we provide its decomposition. Finally, in Section 8 we consider an application to Mathematical Finance concerned with the problem of pricing a Credit Default Swap in an elementary market model. The paper closes with the Appendix where, for the sake of easy reference, we recall the Bayes Theorem, and we prove a slight extension of the so-called innovation lemma (used in Section \[sec:Semimartingale-decomposition\] for the semimartingale decomposition of the information process $\beta$) and, finally, an auxiliary result.

This paper is based on the thesis [@key-10] and the main objective of it is to introduce and study the information process $\beta$ and to provide its basic properties. Other topics of the thesis [@key-10] are the enlargement of the filtration $\mathbb{F}^{\beta}$ with a reference filtration $\mathbb{F}$, the classification of the default time $\tau$ (with respect to the filtration $\mathbb{F}^{\beta}$ and with respect to the enlarged filtration) and further applications to Mathematical Finance.

Preliminaries
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