Increments Are Independent of the Natural Filtration Past

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $X=(X_t)_{t\ge0}$ be a stochastic process on it with independent increments. Suppose there is a real number $c$ such that $X_0=c$ almost surely. Let $(\mathcal{F}^{X}_t)_{t\ge0}$ be the natural filtration of $X$.

Then for all real $0\le s<t$, the $\sigma$-algebras $\sigma(X_t-X_s)$ (generated by the random variable $X_t-X_s$) and $\mathcal{F}^{X}_s$ are independent in the sense of Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras.

Consequently, if $Y$ is any $\mathcal{F}^{X}_s$-measurable random variable (meaning $Y^{-1}(B)\in\mathcal{F}^{X}_s$ for every Borel set $B$), then $X_t-X_s$ and $Y$ are independent.