Natural filtration of a process

For a real process $X$ Adapted and integrable stochastic process, its natural filtration is ${\mathcal{F}}_n^X=\sigma (\{X_k^{-1}(B):0\le k\le n,B\in \mathcal{B}(\mathbb{R})\}),n\in {\mathbb{N}}_0\mathrm{.}$ The generated-sigma-algebra theorem Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal applies because the displayed sets are subsets of $\mathrm{\Omega }$. The ambient $\mathcal{F}$ contains every generator, so minimality gives ${\mathcal{F}}_n^X\subseteq \mathcal{F}$. The generator family at $n$ is contained in that at $n+1$; hence minimality gives ${\mathcal{F}}_n^X\subseteq {\mathcal{F}}_{n+1}^X$.

Every Borel preimage under $X_n$ is a generator at time $n$, so $X$ is adapted. If $({\mathcal{G}}_n)$ is any other filtration making $X$ adapted, then for $k\le n$ every $X_k^{-1}(B)$ belongs to ${\mathcal{G}}_k\subseteq {\mathcal{G}}_n$. Minimality therefore gives ${\mathcal{F}}_n^X\subseteq {\mathcal{G}}_n$. This proves the claimed smallest-filtration property. At time zero the generators come from $X_0$; the resulting sigma-algebra need not be trivial. No completion or arbitrary null-set modification is included, and the defining intersection requires no choice.