Question: 4.9. Let r be the first time that a standard Brownian motion B(t) starting from B(0) = x > 0 reaches zero. Let A be

4.9. Let r be the first time that a standard Brownian motion B(t) starting from B(0) = x > 0 reaches zero. Let A be a positive constant. Show that w(x) = E[e-ATIB(0) = x] = e-'-21*.

Hint: Develop an appropriate differential equation by instituting an infinitesimal first step analysis according to w(x) = E[E{e-ATjB(Ot)}!B(0) = x] = E[e-A°'w(x + AB)].

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