},\, t>0)$. On the other hand, these both processes are not indistinguishable, since the process $\beta$ can hit 0 before $\tau$ (in fact, due to the law of iterated logarithm, this happens uncountably many times $\mathbf{P}$-a.s.).

Roughly speaking we can say that in general, if we can observe only $\beta_{t}$, we are sure that $\tau>t$ whenever $\beta_{t}\neq0$. But, if at time $t$ we observe the event $\{ \beta_{t}=0\}$, the information carried by $\mathcal{F}_{t}^{0}$ may be not sufficient to know whether the event $\{ \tau\le t\}$ has occurred or not, we can only say that it occurred $\mathbf{P}$-a.s.

We now show that the information process $\beta$ is a Markov process with respect to its natural filtration $\mathbb{F}^{0}$.

\[TEO:The-information-process\] The information process $\beta$ is an $\mathbb{F}^{0}$-Markov process: $$\mathbf{E}\left[f\left(\beta_{t+h}\right)|\mathcal{F}_{t}^{0}\right]=\mathbf{E}\left[f\left(\beta_{t+h}\right)|\beta_{t}\right],\quad\mathbf{P}\textrm{-a.s., }t\geq0,\label{eq:statementMK1}$$ for all $h\geq0$ and for every measurable function $f$ which is nonnegative or such that $f(\beta_{t+h})$ is integrable.

For $t=0$ the statement is clear. Let us assume $t>0$. On the set $\{ \tau\leq t\}$ we have $\mathbf{P}$-a.s. $$\mathbf{E}\left[f\left(\beta_{t+h}\right)|\mathcal{F}_{t}^{0}\right]\mathbb{I}_{\left\{ \tau\leq t\right\} } =\mathbf{E}\left[f\left(0\right)\mathbb{I}_{\left\{ \tau\leq t\right\} }|\mathcal{F}_{t}^{0}\right]
  =f\left(0\right)\mathbb{I}_{\left\{ \tau\leq t\right\}}
  =f\left(0\right)\mathbb{I}_{\left\{ \beta_{t}=0\right\}}\,,$$ which is a measurable function with respect to $\sigma(\beta_{t})$ and hence (\[eq:statementMK1\]) is valid on $\{ \tau\leq t\}$ $\mathbf{P}$-a.s. Now we have to 