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

22. Let X(t) = σB(t) + μt, and for given positive constants A and B, let p denote the probability that {X(t), t  0} hits A before it hits −B.

(a) Define the stopping time T 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) − μT)/σ − c2T/2}] = 1

(b) Let c = −2μ/σ, and show that E[exp{−2μX(T)/σ}] = 1

(c) Use part

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

Hint: What are the possible values of exp{−2μX(T)/σ2}?