is a consequence of the fact that $\tau$ is an $\mathbb{F}^{P}$-stopping time and that $\beta_{u}=0$ on $\{\tau\leq t<u\}$. On the set $\{ t<\tau\}$, from Lemma \[lem:AUXlemma\] we have $$\begin{aligned}
\mathbf{E}\left[g\left(\tau,\,\beta_{u}\right)|\mathcal{F}_{t}^{P}\right]\mathbb{I}_{\left\{ t<\tau\right\} } & =\mathbf{E}\left[g\left(\tau,\,\beta_{u}\right)|\beta_{t}\right]\mathbb{I}_{\left\{ t<\tau\right\} }\\
 & =\mathbf{E}\left[g\left(\tau,0\right)\mathbb{I}_{\left\{ t<\tau\leq u\right\} }|\beta_{t}\right]+\mathbf{E}\left[g(\tau,\beta_{u})\mathbb{I}_{\left\{ u<\tau\right\} }|\beta_{t}\right]\,,\end{aligned}$$ $\mathbf{P}$-a.s. We remark that due to Corollary \[LEM:prop\_cond\_exp\_tau\] $$\mathbf{E}\left[g\left(\tau,0\right)\mathbb{I}_{\left\{ t<\tau\leq u\right\} }|\beta_{t}\right]=\int_{\left(t,u\right]}g\left(r,0\right)\phi_{t}\left(r,\beta_{t}\right)dF\left(r\right)\,\mathbb{I}_{\{t<\tau\}}\,.$$ On the other hand, from (\[eq:cond exp ext bb\]), for $t<u<r$, $$\begin{aligned}
\mathbf{E}\left[g\left(r,\beta_{u}^{r}\right)|\beta_{t}^{r}=x\right]&=\int_{\mathbb{R}}g\left(r,y\right)p\left(\frac{r-u}{r-t}\left(u-t\right),\,y,\,\frac{r-u}{r-t}x\right)dy\nonumber\\
&=:G_{t,u}\left(r,x\right)\,. \label{eq:prova3}\end{aligned}$$ It follows from Lemma \[lem:AUXlemma\] and from (\[eq:prova3\]) that, on $\{ t<\tau\}$ $\mathbf{P}$-a.s., $$\begin{aligned}
\mathbf{E}\left[g(\tau,\beta_{u})\mathbb{I}_{\left\{ u<\tau\right\} }|\beta_{t}\right] & =\mathbf{E}\left[\left(\mathbf{E}\left[g\left(r,\beta_{u}^{r}\right)\mathbb{I}_{\left\{ u<r\right\} }|\beta_{t}^{r}\right]\right)_{r=\tau}|\beta_{t}\right]\\
 & =\mathbf{E}\left[\left(G_{t,u}\left(r,\beta_{t}^{r}\right)\right)_{r=\tau}|\beta_{t}\right]\\
 & =\mathbf{E}\left[G_{t,u}\left(\tau,\beta_{t}\right)|\beta_{t}\right]\\
 & =\int_{\left(u,+\infty\right)}G_{t,u}\left(r,\beta_{t}\right)\phi_{t}\left(r,\beta_{t}\right)dF\left(r\right),\end{aligned}$$ where the latter equality follows from Proposition \[cor:Let–be-BIS\].

As an immediate consequence of Theorem \[thm:Let–and\], for $t<u$, we can calculate the conditional expectation of $\beta_{u}$ given $\beta_{t}$ by $$\mathbf{E}\left[\beta_{u}|\mathcal{F}_{t}^{P}\right]=\beta_{t}\int_{\left(u,+\infty\right)}\frac{r-u}{r-t}\,\phi_{t}\left(r,\beta_{t}\right)dF\left(r\right)\mathbb{I}_{\left\{ t<\tau\right\} },\; \mathbf{P}\textrm{-a.s.},$$ and the conditional distribution of $\beta_u$ given $\beta_t$: For $\Gamma\in\mathcal{B}(\mathbb{R})$, $\mathbf{P}$-a.s., $$\begin{gathered}
\mathbf{P}\left(\beta_{u}\in\Gamma|\mathcal{F}_{t}^{P}\right)=\mathbb{I}_{\left\{ 0\in\Gamma\right\}}\,\mathbb{I}_{\left\{ \tau\leq t\right\} } +\mathbb{I}_{\left\{ 0\in\Gamma\right\} } \int_{\left(t,u\right]}\phi_{t}\left(r,\beta_{t}\right)dF\left(r\right)\,\mathbb{I}_{\left\{ \tau\leq t\right\} }\\
+\int_{\left(u,+\infty\right)}\int_{\Gamma}p\big(\frac{r-u}{r-t}\left(u-t\right),\, y,\,\frac{r-u}{r-t}\beta_{t}\big)dy\,\phi_{t}\left(r,\beta_{t}\right)dF\left(r\right)\mathbb{I}_{\left\{ t<\tau\right\} }.\label{eq:prova2}\end{gathered}$$

\[rem:Strong\_Inhom\_MK\] From the factor $\left(r-u\right)/\left(r-t\right)$ in (\[eq:prova2\]) we see that the process $\beta$ cannot be a homogeneous Markov process because $\mathbf{P}(\beta_{u}\in\Gamma|\mathcal{F}_{t}^{P})$ does not depend only on $u-t$ and $(\beta_{t},\Gamma)$.

Markov Property\[sec:The-Markov-property2\]
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In this section we are going to strengthen Theorem \[TEO:The-information-process\] on 