I have a question:
Given two disjoint intervals $[a,b]$ and $[c,d]$, how to prove almost surely we have
$$\sup_{t\in[a,b]}B_t\neq\sup_{t\in[c,d]}B_s$$
where $B$ is a standard brownian motion. I have no idea about this problem. Does someone have an idea? Thansk a lot!
Best Answer
If you write $X_t = B_{a+t}-B_a$ for $0 \le t \le b-a$ and $Y_s =B_{c+s}-B_c$ for $0 \le d-c$, then $(X_t)_{t\in[0,b-a]}$ and $(Y_s)_{s\in[0,d-c]}$ are independent Brownian motions, also independent of $B_a$ and $B_c$. So their suprema $$A = \sup\limits_{t\in [0,b-a]}X_t \text{ and } C = \sup\limits_{s\in [0,d-c]}Y_s$$ are independent random variables, which are also independent of $B_a$ and $B_c$. Now $$\mathbb{P}\left[\sup\limits_{t\in[a,b]} B_t = \sup\limits_{t\in[c,d]} B_t\right] =\mathbb{P}[B_a + A = B_c + C] = \mathbb{P}[A-C = B_c - B_a].$$ This is always $0$ since $B_c-B_a$ is continuous (actually, $N(0,c-a)$-distributed) and independent of $A-C$ (which is also continuous unless both $a=b$ and $c=d$.)