the Markov property of the information process $\beta$. We shall prove that $\beta$ is not only a Markov process with respect to the filtration $\mathbb{F}^0$ (or $\mathbb{F}^P$) but also with respect to $\mathbb{F}^\beta$, the smallest filtration containing $\mathbb{F}^0$ and satisfying the usual conditions. As an important consequence, it turns out that the filtrations $\mathbb{F}^P$ and $\mathbb{F}^\beta$ are equal which amounts to saying that the filtration $\mathbb{F}^P$ is right-continuous. The result is stated in the following theorem.

\[thm:The-process–1\]The process $\beta$ is a Markov process with respect to the filtration $\mathbb{F}^{\beta}$, i.e., $$\mathbf{E}\left[g\left(\beta_{u}\right)|\mathcal{F}_{t}^{\beta}\right]=\mathbf{E}\left[g\left(\beta_{u}\right)|\beta_{t}\right],\;\mathbf{P}\textrm{-a.s.}$$ for all $0\leq t<u$ and all measurable functions $g$ such that $g(\beta_{u})$ is integrable.

The proof is divided into two main parts. In the first one we prove the statement of the above theorem for $t>0$, while in the second part we consider the case $t=0$. Throughout the proof we can assume without loss of generality that the function $g$ is continuous and bounded by some constant $M\in\mathbb{R}_+$. For the first part of the proof, let $t>0$ be a strictly positive real number and let $(t_{n})_{n\in\mathbb{N}}$ be a decreasing sequence converging to $t$ from above: $0<t<...<t_{n+1}<t_{n}<u,\: t_{n}\downarrow t$ as $n\rightarrow\infty$. From the definition of $\mathcal{F}_{t}^{\beta}$ we have that $\mathcal{F}_{t}^{\beta}=\bigcap_{n\in\mathbb{N}}\mathcal{F}_{t_{n}}^{P}$ 