at which the bankruptcy of a company (or state) occurs. The time at which this crucial event takes place is called *default time* and it is modeled by a strictly positive random variable $\tau$. We tackle the problem by defining the process $\beta=(\beta_{t},\, t\geq0)$, a Brownian bridge between 0 and 0 on the random time interval $[0,\tau]$: $$\beta_{t}:=W_{t}-\frac{t}{\tau\vee t}W_{\tau\vee t},\quad t\geq 0\,, \label{eq:EQ0}$$ where $W=(W_{t},\, t\geq0)$ is a Brownian motion independent of $\tau$. Since we are going to model the information about $\tau$ with such a bridge process, we call $\beta$ the *information process*. The filtration $\mathbb{F}^{\beta}=(\mathcal{F}_{t}^{\beta})_{t\geq0}$ generated by the information process provides partial information on the default time before it occurs. The intuitive idea is that when market agents observe $\beta_{t}$ away from 0, they know that the default will not occur immediately; on the other hand, the period of fear of an imminent default corresponds to the situation in which $\beta_{t}$ is close to 0. In our approach the Bayes formula is of particular importance since it allows to infer the a posteriori probability distribution of the default time $\tau$ given the information carried by $\mathcal{F}_{t}^{\beta}$, for $t\geq0$.

Two classes of credit risk models have greatly inspired our 