I was reading the Wikipedia article about the reflection principle for Wiener processes, which can be written as :
Lemma (Reflection Principle) : If $(W(t) : t \ge 0)$ is a Wiener process and $a > 0$ is a threshold, then $\forall t\ge 0$ $$ \mathbb{P}\left(\sup_{0\le s\le t} W(s) \ge a \right) = 2\mathbb{P}(W(t) \ge a)$$
And there are some parts of the proof that I can't seem to understand. After defining $\tau_a := \inf \{t : W(t) \ge a\}$ and $X(t) := W(t + \tau_a) - a$, the following points are unclear :
- The proof relies on the fact that events $\{\sup_{0\le s\le t} W(s) \ge a \}$ and $\{X(t - \tau_a) < 0\}$ are independent conditional on $\mathcal{F}_{\tau_a}^W$. I understand that by the strong Markov property, $X(t)$ is a Brownian Motion independent of $\mathcal{F}_{\tau_a}^W$, but to me that only holds for $t \ge 0$. However as far as I know there are no reasons for $ t - \tau_a $ to be nonnegative.
- The point above raises a second concern : if $ t - \tau_a < 0 $, then $X(t - \tau_a)$ doesn't make sense if you think of $X$ as a Brownian Motion, since they are only defined for nonnegative times. I thus fail to understand how the probability $\mathbb{P}\left(\sup_{0\le s\le t} W(s) \ge a, X(t - \tau_a)\right) $ is well defined.
Any help on this would be greatly appreciated.
Bonus question (if this is inappropriate I will create a separate question) : Why is the lemma as written above equivalent to $\mathbb{P}(\tau_a \leq t) = 2 \mathbb{P}(W(t) \geq a)$ ?
I think answering the bonus question likely answers the two main questions as well. The reason is that $\{\tau_a \le t \} = \{ \sup_{0 \le s \le t} W(s) \ge a\}$, i.e. $W$ hits $a$ at a time before $t$ if and only if the max up to time $t$ is at least $a$. Therefore on the event $\{ \sup_{0 \le s \le t} W(s) \ge a\}$ we have that $t \ge \tau_a$, so $t - \tau_a \ge 0$ and $X(t-\tau_a)$ is well defined on that event.