0

Let $W_t$ be a standard Brownian motion starting in zero, $a, b \in \Bbb R$. Let $T_{a, b}$ denote the first hitting time of the straight line $s\mapsto a +bs$, i.e $$ T_{a, b} = \inf \Bigl\{t \geq 0: W_t = a +bt \Bigr\}. $$

My question: is it true that $$ T_{a, b} \stackrel{d}{=} T_{-a, -b}? $$ From an intuitive point this seems to be true, however I am blanking on giving a formal proof.

My goal is to calculate $\Bbb P(T_{a,b} \leq t) $ for $a < 0$, similar to here .

Cettt
  • 2,494

1 Answers1

1

Hint:

$$T_{a,b}=\inf\{t \geq 0; W_t = a+bt\} \stackrel{d}{=} \inf\{t \geq 0; B_t = a+bt\}$$

for any two Brownian motions $(W_t)_{t \geq 0}$ and $(B_t)_{t \geq 0}$. How to choose $(B_t)_{t \geq 0}$ such that the right-hand side equals $T_{-a,-b}$?

saz
  • 120,083