that $g(\tau,\beta_{t})$ is integrable. Then, $\mathbf{P}$-a.s., $$\mathbf{E}\left[g\left(\tau,\beta_{t}\right)|\mathcal{F}_{t}^{\beta}\right]=g\left(\tau,0\right)\mathbb{I}_{\left\{ \tau\leq t\right\} }+\int_{\left(t,+\infty\right)}g\left(r,\beta_{t}\right)\phi_{t}\left(r,\beta_{t}\right)dF\left(r\right)\mathbb{I}_{\left\{ t<\tau\right\} }\,.$$

The statement is clear on the set $\{ \tau\leq t\} $. On the set $\{ t<\tau\}$, we first prove it for bounded measurable functions $g$ and, by a monotone class argument, it suffices to consider functions $g$ of the form $g(\tau,\beta_{t})=g_{1}(\tau)g_{2}(\beta_{t})$ where $g_1$ and $g_2$ are bounded measurable functions. Then we have that $$\begin{aligned}
\mathbf{E}\left[g\left(\tau,\beta_{t}\right)|\mathcal{F}_{t}^{\beta}\right]\mathbb{I}_{\left\{ t<\tau\right\} } & =\frac{\int_{\left(t,+\infty\right)}g_{1}\left(r\right)\varphi_{t}\left(r,\beta_{t}\right)dF(r)}{\int_{\left(t,+\infty\right)}\varphi_{t}\left(v,\beta_{t}\right)dF(v)}\mathbb{I}_{\left\{ t<\tau\right\} }\,g_{2}\left(\beta_{t}\right)\\
 & =\int_{\left(t,+\infty\right)}g\left(r,\beta_{t}\right)\phi_{t}\left(r,\beta_{t}\right)dF\left(r\right)\mathbb{I}_{\left\{ t<\tau\right\} }.\end{aligned}$$ If $g$ is a nonnegative measurable function, we can apply the above result to the functions $g_N:=g\wedge N$ and using the monotone convergence theorem we obtain the asserted equality for $g$. Finally, in the general case, the result is true for the positive and negative parts $g^+$ and $g^-$ of $g$ and for $g=g^+-g^-$ the equality follows from the linearity of both sides.

We can now state the main result of this section.

\[thm:Let–and\] Let $0<t<u$ and $g$ be a measurable function defined on $(0,+\infty)\times\mathbb{R}$ such that $\mathbf{E}\left[|g(\tau,\,\beta_{u})|\right]<+\infty$. Then, $\mathbf{P}$-a.s. $$\begin{aligned}
\lefteqn{\mathbf{E}\left[g\left(\tau,\,\beta_{u}\right)|\mathcal{F}_{t}^{P}\right]=\mathbf{E}\left[g\left(\tau,\,\beta_{u}\right)|\beta_t\right]}\\
=&\;g\left(\tau,0\right)\mathbb{I}_{\left\{ \tau\leq t\right\} }+\int_{\left(t,u\right]}g\left(r,0\right)\phi_{t}\left(r,\beta_{t}\right)dF\left(r\right)\,\mathbb{I}_{\left\{ t<\tau\right\}}\\
&+\!\!\int\limits_{\left(u,+\infty\right)}\!\!\int\limits_{\mathbb{R}}g\left(r,y\right)p\big(\frac{r-u}{r-t}\left(u-t\right),\, y,\,\frac{r-u}{r-t}\beta_{t}\big)dy\phi_{t}\left(r,\beta_{t}\right)dF\left(r\right)\mathbb{I}_{\left\{ t<\tau\right\} }\,, \label{eq:prova1}\end{aligned}$$ where $p(t,\cdot,y)$ is the Gaussian density with mean $y$ and variance $t$.

On the set $\{ \tau\leq t\} $ the statement 