\ t\geq 0)$ for updating the initial knowledge on $\tau$. At time 0, the market agents have only *a priori* knowledge about $\tau$, represented by its distribution function $F$. As time is increasing, information concerning the default becomes available. Using the Bayes theorem (recalled in the Appendix for easy reference), the *a posteriori* distribution of $\tau$ based on the observation of $\beta$ up to $t$ can be derived and in this way agents can update their initial knowledge obtaining a sharper estimate of the default time $\tau$.

In this section the $\sigma$-algebra generated by the future of $\beta$ at time $t$ is denoted by $\mathcal{F}^P_{t,\infty}:=\sigma(\beta_{s},\, t\leq s\leq+\infty)\vee\mathcal{N}_{P}$. The following is a standard result on Markov processes:

\[lem:basic MKV\]Let $X=(X_{t},\, t\geq0)$ be a stochastic process adapted to a filtration $\mathbb{F}=(\mathcal{F}_{t})_{t\geq0}$. Then the following are equivalent:

1.  $X$ is Markov with respect to $\mathbb{F}$.

2.  For each $t\geq0$ and bounded (or nonnegative) $\sigma(X_{s},\, s\geq t)$-measurable random variable $Y$ one has $$\mathbf{E}\left[Y|\mathcal{F}_{t}\right]=\mathbf{E}\left[Y|X_{t}\right], \; \mathbf{P}\textrm{-a.s.}$$

See [@key-5 Ch. I, Theorem (1.3)].

The next proposition describes the structure of the a posteriori distribution of $\tau$ based on the observation of $\mathcal{F}^P_t$.

\[prop:HJ1\] For all $t,u\geq0$, it holds $$\label{prop:HJ1second}
\mathbf{P}\left(\tau\leq u|\mathcal{F}_{t}^{P}\right)=\mathbb{I}_{\left\{ \tau\leq t\wedge u\right\} }+\mathbf{P}\left(t<\tau\leq u|\beta_{t}\right)\mathbb{I}_{\left\{ 