Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$, $E$ be a $\mathbb R$-Banach space and $(X_t)_{t\ge0}$ be an $E$-valued process on $(\Omega,\mathcal A,\operatorname P)$.
I would like to show that
- $X_{s+t}-X_s$ and $\mathcal F_s$ are independent for all $s,t\ge0$
- $\left(X_{t_i}-X_{t_{i-1}}\right)_{1\le i\le k}$ is independent for all $k\in\mathbb N$ and $0\le t_0<\cdots<t_k$
are equivalent.
My idea for "(1.) $\Rightarrow$ (2.)" is the following: We first show the following claim: $X_{t_i}-X_{t_{i-1}}$ and $\left(X_{t_1}-X_{t_0},\ldots,X_{t_{i-1}}-X_{t_{i-2}}\right)$ are independent for all $i\in\{2,\ldots,k\}$:
- Let $i\in\{2,\ldots,k\}$ and $$f:E^i\to E^i\;,\;\;\;x\mapsto(x_1,x_2-x_1,\ldots,x_i-x_{i-1})$$
- $f$ is clearly Borel measurable(, linear) and injective. Thus, if we were able to show that $$f(B)\in\mathcal B(E^i)\;\;\;\text{for all }B\in\mathcal B(E^i)\tag1,$$ it is a well-known result that \begin{equation}\begin{split}\sigma\left(X_{t_0},X_{t_1}-X_{t_0},\ldots,X_{t_k}-X_{t_{k-1}}\right)&=\sigma\left(f\left(X_{t_0},\ldots,X_{t_k}\right)\right)\\&=\sigma\left(X_{t_0},\ldots,X_{t_k}\right)\subseteq\mathcal F_{t_{i-1}}\end{split}\tag2\end{equation}
- By (1.), $X_{t_i}-X_{t_{i-1}}$ is independent of $\mathcal F_{t_{i-1}}$ and hence, by $(2)$, independent of $X_{t_0},X_{t_1}-X_{t_0},\ldots,X_{t_k}-X_{t_{k-1}}$.
Can we conclude the first direction from this claim? And how can we show $(1)$? And do we need to assume that $X$ is $\mathcal F$-adapted to show the other direction?
