# Brownian Motion Independent Increments

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?

• Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be searched and are not view-able to some, such as those who use screen readers. I have edited your question to reflect this principle. Feb 25, 2020 at 3:50

$$\mathcal{F}_s^X=\left\{ \left\{(X_{t_1}, X_{t_2}, ... )\in B\right\}\:|\: 0\leq t_1 < t_2 < ... \leq s,\: B\in \mathcal{B}(\mathbf{R})\otimes \mathcal{B}(\mathbf{R})\otimes ... \right\}$$,
$$\left\{ X_{t_1}\in B_1, X_{t_2}\in B_2, ...\:|\: 0\leq t_1 < t_2 < ... \leq s,\: B_n\in \mathcal{B}(\mathbf{R})\right\}$$