Let $ X(t): t ≥ 0 % be a Brownian motion with variance parameter σ 2 . Let T α be the time of hitting α first. Let Y ∼ N (0, σ 2 /α 2 ). Show that, for α > 0, T α and 1/Y 2 are identically...

Let $ X(t): t ≥ 0 % be a Brownian motion with variance parameter σ². Let T_α
be the time of hitting α first. Let Y ∼ N (0, σ²/α²). Show that, for α > 0, T_α
and 1/Y²
are identically distributed.

May 13, 2022