Problem 33
(Hitting Times for Brownian Motion) Let W(t) be a standard Brownian motion. Let a > 0. 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
fTa(t) = a/(t∑(2πt)) exp{-a^2/2t}.
Note: By symmetry of Brownian motion, we conclude that for any a ≠ 0, we have
fTa(t) = |a|/(t∑(2πt)) exp{-a^2/2t}.