random time with distribution function denoted by $F$: $F(t):=\mathbf{P}(\tau\leq t),\: t\geq0$. In this paper the following assumption will always be made.

\[ass:assumption1\]The random time $\tau$ and the Brownian motion $W$ are independent.

The random time $\tau$ (which will be interpreted as *default time*) is supposed to be not an $\mathbb{F}^{W}$-stopping time. This means that the case in which $\tau$ is equal to a positive constant $T$ will not be considered.

\[def:obs by MonPon\]The process $\beta=(\beta_{t},\;t\geq0)$ given by $$\beta_{t}:=W_{t}-\frac{t}{\tau\vee t}W_{\tau\vee t},\; t\geq0,\label{eq:betaDEF1}$$ will be called *information process*.

In what follows, we shall use the following filtrations:

1.  $\mathbb{F}^{0}=(\mathcal{F}_{t}^{0})_{t\geq0}$, the natural filtration of the process $\beta$:\
    $\mathcal{F}_{t}^{0}:=\sigma(\beta_{s},\,0\leq s\leq t),\ t\geq0$.

2.  $\mathbb{F}^{P}=(\mathcal{F}_{t}^{P})_{t\geq0}$, the completed natural filtration:\
    $\mathcal{F}_{t}^{P}:=\mathcal{F}_{t}^{0}\vee\mathcal{N}_{P},\, t\geq0$.

3.  $\mathbb{F}^{\beta}=(\mathcal{F}_{t}^{\beta})_{t\geq0}$, the smallest filtration containing $\mathbb{F}^{0}$ and satisfying the usual hypotheses of right-continuity and completeness: $$\mathcal{F}_{t}^{\beta}:=\sigma\left(\bigcap_{u>t}\mathcal{F}_{u}^{0}\cup\mathcal{N}_{P}\right)=\mathcal{F}_{t+}^{0}\vee\mathcal{N}_{P},\: t\geq0.$$

The aim of this section is to prove that the default time $\tau$ is a stopping time with respect to the completed natural filtration $\mathbb{F}^P$ and that the information process $\beta$ is Markov with respect to $\mathbb{F}^P$.

Although it is possible to prove directly that $\tau$ is an $\mathbb{F}^{\beta}$-stopping time (see [@key-10 Lemma 2.5]) we point out the following result which better involves the role of 