total probability, Corollary \[cor:LEM if–is\] and , we can calculate $$\begin{aligned}
\mathbf{E}\left[g\left(\beta_{u}\right)|\beta_{0}\right]&=\mathbf{E}\left[g\left(\beta_{u}\right)\right]\\
&=g\left(0\right)\,F\left(u\right)+\int_{\left(u,+\infty\right)}\int_{\mathbb{R}}g\left(y\right)p\left(\frac{r-u}{r}u,y,0\right)dy\,dF\left(r\right).\end{aligned}$$ Consequently, recalling (\[eq:equiv1\]) for computing the left-hand side of (\[eq:equiv 2\]) and the definition of $G_{t_n,u}$ and $G_{t,u}$ by and noting the obvious relation $\lim_{n\rightarrow\infty}\mathbb{I}_{\{ \tau\leq t_{n}\} }=0$, it is sufficient to prove the following two equalities, $\mathbf{P}$-a.s.: $$\begin{aligned}
\lim_{n\rightarrow\infty}\int_{\left(t_{n},u\right]}\phi_{t_{n}}\left(r,\beta_{t_{n}}\right)dF\left(r\right) &=F\left(u\right)\,,\label{eq:star 2 bis}\\
\lim_{n\rightarrow\infty}\int\limits_{\left(u,+\infty\right)}G_{t_n,u}\left(r,\beta_{t_n}\right)\,\phi_{t_{n}}\left(r,\beta_{t_{n}}\right)dF\left(r\right)&=\int\limits_{\left(u,+\infty\right)}G_{t,u}\left(r,\beta_t\right)\,dF\left(r\right)\,.\label{eq:star 3 bis}\end{aligned}$$ The first equality relies on Lemma \[lem:On-the-set-2\] and Lebesgue’s bounded convergence theorem. For verifying the second equality we use Lebesgue’s bounded convergence theorem combined with Lemma \[lem:On-the-set-2\] and the boundedness and pointwise convergence of the sequence $G_{t_n,u}(\cdot,\beta_{t_n})$ (see (\[eq:HJ1\]) and (\[eq:HJ2\])).

We now turn to the general case where $\tau>0$. Let $\varepsilon>0$ be arbitrary, but fixed. In what follows the process $^{\varepsilon}\!\beta=(^{\varepsilon}\!\beta_{t},\, t\geq0)$ will denote the process defined by $^{\varepsilon}\!\beta_{t}:=
(\beta_{t}^{r})_{r=\tau\vee\varepsilon}$. The natural filtration of the process $^{\varepsilon}\!\beta$ will be denoted by $\mathbb{F}^\varepsilon=(\mathcal{F}^\varepsilon_{t})_{t\geq0}$, where $\mathcal{F}_{t}^\varepsilon :=\sigma(^{\varepsilon}\!\beta_{s},\,0\leq s\leq t)$. Obviously, for proving the Markov property of $\beta$ with respect to $\mathbb{F}^\beta$ at $t=0$ it is sufficient to show that $\mathcal{F}^\beta_0:=\mathcal{F}^0_{0+}\vee\mathcal{N}_P$ is $\mathbf{P}$-trivial. In order to show that $\mathcal{F}_{0+}^{0}\vee\mathcal{N}_{P}$ is indeed the trivial $\sigma$-algebra, we consider a set $A\in\mathcal{F}_{0+}^{0}$ and we show that if $\mathbf{P}(A)>0$, then $\mathbf{P}(A)=1$.

If $\mathbf{P}(A)>0$, then there exists an 