Given a standard Brownian motion $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P},(B_t)_t)$ (the standard filtration $(\mathcal{F}_t)_t$), we define
$$\forall t\ge 0: M_t:=\max_{0\le s\le t} B_s$$
Our professor presented to us the following Proposition with the name of Reflection Principle:
- $\mathbb{P}(M_t\ge a)=2\mathbb{P}(B_t\ge a)$ for all $a\ge0$, for all $t\ge 0$
I don't understand the interpretation of this result.
The result is that the maximum of a standard Brownian motion has the same distribution as the absolute value of the Brownian motion. The proof follows from the property that the reflected Brownian motion $M_t-B_t$ has the same properties as the Brownian motion process itself.