3
$\begingroup$

Suppose that $B_t$ is a standard Brownian motion. And $T_a$, $T_b$ are the hitting time whereas $a<0$, $b>0$. Then are these two random variables independent?

1 Answers 1

3

No, hitting times for the same Brownian Motion are not independent. Think of it this way: given $T_a = s$, $B_s = a$ so if $a \ne b$, $B_t$ is much less likely to be close to $b$ when $t$ is near $s$, and therefore $T_b$ is much less likely to be close to $s$.

  • 0
    I see. Thanks very much!2011-09-28
  • 0
    May I know what's the probability of T_b < T_a2012-05-18
  • 0
    if $c=\mathbb{P}(T_b$0=\mathbb{E}[B_{T_a\wedge T_b}]=a(1-c)+bc$. Solve for $c$. – 2012-05-18