Let W(t) be a standard Brownian motion. Let a > 0. Define Ta as the first time that W(t) = a. That is

a. Show that for any t ≥ 0, we have

b. Using Part (a), show that

c. Using Part (b), show that the PDF of T_a is given by

By symmetry of Brownian motion, we conclude that for any a ≠ 0, we have

Transcribed Image Text:

Ta = min{t: W(t) = a}.