t<\tau\right\} },\;\mathbf{P}\textrm{-a.s.}$$

Obviously, we have $\{ \tau\leq u\} =\{ \tau\leq t\wedge u\} \cup\{ t<\tau\leq u\}$. The first set of the right-hand side of the above decomposition yields the first summand of the statement. Then it suffices to observe that we have the relation $\{ t<\tau\leq u\} =\{ \beta_{t}\neq0,\beta_{u}=0\}$ $\mathbf{P}\textrm{-a.s.}$ where the set on the right-hand side of the above equality belongs to $\mathcal{F}^P_{t,\infty}$. It remains to apply Lemma \[lem:basic MKV\] in order to complete the proof of the statement.

Recalling that $F$ denotes the distribution function of $\tau$ and formula (\[eq:bbdensity\]) for the definition of the function $\varphi_{t}(r,x)$ (which is equal to the density of the Brownian bridge $\beta_{t}^{r}$ at time $t<r$), we have the following result which provides the explicit form of the a posteriori distribution of $\tau$ based on the observation of $\beta$ up to $t$.

\[lem:PRE-theo-4.3\]Let $t>0$. Then, for all $u>0$, $\mathbf{P}$-a.s. $$\begin{aligned}
\mathbf{P}\left(\tau\leq u|\mathcal{F}_{t}^{P}\right) =\mathbb{I}_{\left\{\tau\leq t\wedge u\right\}}+\frac{{\displaystyle \int_{\left(t,u\right]}}\varphi_{t}\left(r,\beta_{t}\right)dF(r)}{{\displaystyle \int_{\left(t,+\infty\right)}}\varphi_{t}\left(v,\beta_{t}\right)dF(v)}\,\mathbb{I}_{\left\{ t<\tau\right\}}\,.\label{eq:condexptau-1}\end{aligned}$$

The result is a consequence of Proposition \[prop:HJ1\] and the Bayes formula (see Corollary \[cor:(Bayes-formula)-a.s.\]).

Theorem \[lem:PRE-theo-4.3\] can be extended to functions $g$ on $\mathbb{R}_+$ as it will be stated in the following corollary.

\[LEM:prop\_cond\_exp\_tau\]Let $t>0,\; g:\mathbb{R}_{+}\rightarrow\mathbb{R}$ be a Borel function such that $\mathbf{E}\left[|g(\tau)|\right]<+\infty$. Then, $\mathbf{P}$-a.s., 