104 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES
- Prove: In a random walk starting ata >0 the probability to reach the
origin before returning to the starting point equalsqqa− 1.
- In the simple casep= 1/2 =q, conclude from the preceding problem:
In a random walk starting at the origin, the number of visits to the
pointa >0 that take place before the first return to the origin has
a geometric distribution with ratio 1−qqa− 1. (Why is the condition
q≥pnecessary?)
- (a) Draw a sample path of a random walk (withp= 1/2 =q) starting
from the origin where the walk visits the position 5 twice before
returning to the origin.
(b) Using the results from the previous problems, it can be shown
with careful but elementary reasoning that the number of times
N that a random walk (p= 1/2 =q) reaches the valueaa total
ofntimes before returning to the origin is a geometric random
variable with probability
P[N=n] =
(
1
2 a
)n(
1 −
1
2 a
)
.
Compute the expected number of visitsE[N] to levela.
(c) Compare the expected number of visits of a random walk (p=
1 /2 =q) to the value “1 million” before returning to the origin
and to the level 10 before returning to the origin.
- This problem is adapted fromStochastic Calculus and Financial Ap-
plicationsby J. Michael Steele, Springer, New York, 2001, Chapter 1,
Section 1.6, page 9. Information on buy-backs is adapted from investor-
words.com. This problem suggests how results on biased random walks
can be worked into more realistic models.
Consider a naive model for a stock that has a support level of $20/share
because of a corporate buy-back program. (This means the company
will buy back stock if shares dip below $20 per share. In the case
of stocks, this reduces the number of shares outstanding, giving each
remaining shareholder a larger percentage ownership of the company.
This is usually considered a sign that the company’s management is
optimistic about the future and believes that the current share price is
