Let X(t) = o B(t) + gt, and peak given positive constants An and B, let p mean the likelihood {X(t), t

2 0} that hits A preceding it hits - B.

(a) Define the halting time T to be the first run through the interaction nits either An or 8. Utilize this

halting tme and the Martingale characterized in Exercise 19 to snow that

- c2T/2}] = 1

(b) Let c - 2/3/0, and snow that

(c) Use part (b) and the definition ot T to discover p.

Clue: What are the potential qualities otexp{

question 38

Let {X(t), t 2 0} be Brownian movement witn float coefficient u and change boundary 02. Assume

that u > O. Let x > O and characterize the halting time T (as in Exercise 21) by

T - Ming: X(t) = x)

utilize the Martingale characterized in Exercise 18, along witn the outcome ot Exercise 21, to snow that

Var(T) = x021g3

In Exercises 25 to 27, {X(t), t 2 0) is a Brownian movement measure witn float boundary g and

change boundary 02 _

question 39

Let {X(t),

(a) Show that

x} be a pitifully fixed interaction having covariance work Rx(s)

= covpqt),

vareqt + s), - X(t)) = 2Rx(O) - Rx(t).

(b) ltY(t) (X(t+ 1), X(t) snow that {Y(t), < t < x} is additionally feebly fixed having a

covariance work Ry(s) Cov[Y(t), t + s)] that fulfills

RAS-I) Rx(S+ 1)

question 40

Let {X(t), < t < x} be feebly fixed witn covariance work R(s) = Cov(X(t), X(t+ s)) and

let (u,) indicate the force otherworldly thickness of the cycle.

(I) snow that (u,) = R ( It can be shown that

1

(ii) utilize the former to snow that

(w) dw = 27T