among many other works on the reduced-form approach to credit-risk. Besides the advantages that have made this approach well-known, there is the poor structure of the information concerning $\tau$: people just know if the default has occurred ($H_{t}=1$) or not ($H_{t}=0$). Financial reality is often more complex than this: market agents have indeed more information and there are actually periods in which the default is more likely to happen than in others. We try to reconcile those two approaches considering that in our model the information is carried by the process $\beta$ given by (\[eq:EQ0\]).

This paper is organized as follows. In Section 2 we provide some preliminary facts that are used throughout the paper. In Section 3 we give the definition of a bridge $\beta$ of random length $\tau$. We show that $\tau$ is a stopping time with respect to the natural, completed filtration generated by the process $\beta$. Also, we establish that $\beta$ is a Markov process with respect to the filtration $\mathbb{F}^{0}:=(\sigma(\beta_{s},\,0\leq s\leq t))_{t\geq0}$. In Section 4 we derive Bayesian estimates of the distribution of the default time $\tau$. Thereafter, in Section 5 we extend the results obtained in the previous section to more general situations. Section 