22. Let X(t) = () + , and for given positive constants A and B, let ...

22. Let X(t) = óÂ(ß) + ìß, and for given positive constants A and B, let

ñ denote the probability that {X(t), ß > 0} hits A before it hits -B.

(a) Define the stopping time Ô to be the first time the process hits either A or -B. Use this stopping time and the Martingale defined in Exercise 19 to show that E[exp[c(X(T) - ìÔ)/ó - c2T/2}] = 1

(b) Let c = - 2ì/ó, and show that

Å[å÷ñ{-2ì×(Ô)/ó2}] = 1

(c) Use part

(b) and the definition of Ô to find p.

Hint: What are the possible values of å÷ñ{-2ì×(Ô)/ó2}º