Canonical Filtration of a partial sum

Question

I wonder if the canonical filtration of a partial sum of a discrete stochastic process is equal to the canonical filtration of the process it self e.g. Is ${\sigma}(X_1,\ldots,X_n)={\sigma}(X_1,X_1+X_2,\ldots,X_1+\ldots+X_n)$ where $(X_n)_{n\in \mathbb{N}}$ is a discrete real valued stochastic process.

Clearly we have ${\sigma}(X_1,\ldots,X_n)\supseteq{\sigma}(X_1,X_1+X_2,\ldots,X_1+\ldots+X_n)$ but what about the other inklusion?

Answer 1

Yes, the reverse inclusion also holds. If we define $S_n = \sum_{k=1}^n X_k$ then $X_n = S_n-S_{n-1}$, from which we conclude that $\sigma(S_1,...,S_n) \supseteq \sigma(X_1,...,X_n)$.