In Ch 4.2 of Evans's "An Introduction to Stochastic Differential Equations", I have a question.
Let
- $W(\cdot)$ be a one-dimensional Brownian motion defined on some probability space $(\Omega, \mathcal{U}, P)$
- $\mathcal{W}(t) := \mathcal{U}(W(s) \vert 0\le s \le t)$
- $\mathcal{W}^+(t) := \mathcal{U}(W(s) - W(t) \vert s \ge t)$
and
- $\mathcal{F}(t) := \mathcal{U}(W(s) (0 \le s \le t), X_0) $, where $X_0 $ is a random variable independent of $\mathcal{W}^+(0)$.
Then, I am wondering why $\mathcal{F}(t)$ is indepedent of $\mathcal{W}^+(t)$ for all $t \ge 0$. When $t= 0$, it is trivial. However, if $t > 0$, a problem arises because of $X_0$. $\mathcal{W}(t)$ and $\mathcal{W}^+(t)$ are independent since $W(t)$ is a Browinian motion, but $\mathcal{F}(t)$ is finer than $\mathcal{W}(t)$. Any help would be appreciated!
$X_0$ is independant of $\mathcal{W}^+(0)$ and is independant of $\mathcal{W}^+(t)$ as a result (by inclusion, the future since the start of time contains the information of the future since a time t>0). Then since you know $\mathcal{W}^+(t)$ and $\mathcal{W}(t)$ are independant you conclude that $\mathcal{F}(t)$ is independant of $\mathcal{W}^+(t)$ since it is generated by tribes that are independant of $\mathcal{W}^+(t)$ by the coalition lemma.