(Hitting Times for Brownian Motion) Let W(t) be a standard Brownian motion. Let a > 0 and define Ta as the first time that W(t) = a. That is, Ta = min{t : W(t) = a}.
a. Show that for any t ≥ 0, we have P(W(t) ≥ a) = P(W(t) ≥ a | Ta ≤ t)P(Ta ≤ t).
b. Using Part (a), show that P(Ta ≤ t) = 2 [1 - Φ(a/∙t)].
c. Using Part (b), show that the PDF of Ta is given by f_{Ta}(t) = (a / (t√(2πt))) * exp(-a^2 / (2t)).
Note: By symmetry of Brownian motion, we conclude that for any a ≠ 0, we have f_{Ta}(t) = (|a| / (t√(2πt))) * exp(-a^2 / (2t)).