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 $r \leq s \leq t$?
2026-03-25 23:39:19.1774481959
Independent increments and independence of natural filtration
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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, $r_1<r_2<\cdots<r_N\leq r$ and each $A_i$ is a Borel set.