I want to prove the following result but I'm not sure how to go about it:
Let $X=\left\{ X_t \,\vert\,0\leq t<\infty\right\}$ be a stochastic process for which $X_0,X_{t_1}-X_{t_0},\ldots,X_{t_n}-X_{t_{n-1}}$ are independent random variables, for every integer $n\geq 1$ and indices $0=t_0 < t_1 < \cdots < t_n < \infty$. Then for any fixed $0\leq s < t <\infty$, the increment $X_t -X_s$ is independent of $\mathscr F_s^X$.
The statement means trivially that $X_t - X_s$ is independent of $\sigma(X_k)$ for all $0 \leq k \leq s$ and is therefore independent of $\bigcup\limits_{0 \leq k \leq s} \sigma(X_k)$.
However, this is not necessarily a $\pi$ system so I can't necessarily apply Dynkin's theorem to state that $\mathcal{F}_s^X \equiv \sigma\left(\bigcup\limits_{0 \leq k \leq s} \sigma(X_k)\right)$ is independent of $X_t - X_s$.
Does anyone know any generating $\pi$ systems for which $X_t-X_s$ is independent?