What are the generating sets of $\mathcal{F}_t^X=\sigma(X_s , 0 \leq s \leq t) $?

Question

When can $\mathcal{F}_t^X=\sigma(X_s , 0 \leq s \leq t) $ be generated by sets of the form $F=\{X_{t_0} \in \Gamma_1\, X_{t_1} \in \Gamma_2 ,\dots, X_{t_n} \in \Gamma_n \}$ where $\Gamma \in \mathcal{B}(\mathbb{R}^n)$ and $0=t_0<t_1 <t_2 \dots <t_n=t$ ??

I mean under what regularity conditions on the stochastic process $X$ is the above true?

I suspect that right or left continuity(can't even show this) of $X$ implies the existence of such generating sets? Would this be true for any Markov Process $X$.

Any help would be appreciated.

Answer 1

No condition! Just take $\sigma \{X_{t_0}\in \Gamma_1,.....,X_{t_n}\in \Gamma_n\}$ and verify that this is equal to $\sigma \{X_s:0\leq s\leq t\}$ by verifying that each side is contained in the other.