# Independent increments and independence of natural filtration

I am studying a stochastic process $$(X_t)$$ in continuous time which has independent increments. For $$\mathscr{F}_t$$ being the natural filtration I would like if $$\mathop{\mathbb{E}}[X_t-X_s \vert \mathscr{F}_s] = \mathop{\mathbb{E}}[X_t-X_s]$$ whenever $$s \leq t$$. Is it true and do we in general have that $$X_t-X_s$$ is independent of $$\mathscr{F}_r$$ for $$r \leq s \leq t$$?

Yes, it is true. You can prove this by showing that $$(X_t-X_s)^{-1}(A)$$ is independent of $$X_{r_1}^{-1}(A_1) \cap X_{r_2}^{-1}(A_2) ... \cap X_{r_N}^{-1}(A_N)$$ whenever $$N$$ is a positive integer, $$r_1 and each $$A_i$$ is a Borel set.

• Thank you. What you are hinting at here is it to use that $\mathscr{F}_r$ is generated by intersections on the form $X_{r_1}^{-1}(A_1) \cap ... \cap X_{r_N}^{-1}(A_N)$ as you write and the use that it is enough to show independence of a generator of the sigma algebra? If so, can you be a bit more explicit on how to continue the proof from here? It will be appreciated a lot. – J. Goles Sep 3 '19 at 6:45
• @J.Goles One way is to use the $\pi -\lambda$ theorem. Consider $\{ F: \in \mathcal F_r: P( E\cap F)=P(E)P(F)$ where $E=(X_t-X_s)^{-1}(A)$. This is a $\lambda system and we have proved that it contains a$\pi$system which generates$\mathcal F_r\$. – Kavi Rama Murthy Sep 3 '19 at 7:34
• Thank you. I will look further in to that. – J. Goles Sep 9 '19 at 6:12