This preview shows pages 1–2. Sign up to view the full content.
IEOR 4106, Spring 2011, Professor Whitt
Topics for Discussion: Tuesday, April 19
Introduction to Brownian Motion, Chapter 10, Sections 10.110.3
We follow the Ross text in Sections 10.110.3 closely in this opening lecture on Brownian
motion
1 Section 10.1 Basic Properties
1. A Random Walk with Frequent Small Steps
Brownian motion can be thought of as a random walk with frequent small steps. As
usual with random walks, the steps are random, being independent and identically distributed
(i.i.d.). The random walk can be a simple random walk that goes either up or down a constant
amount, each with probability 1
/
2. However, let the time between steps by Δ
t
and let the size
of the step be Δ
x
. Then, at each step, the random walk goes up Δ
x
or down Δ
x
, each with
probability 1
/
2. But the time taken for each step is Δ
t
.
The idea is to let Δ
t
and Δ
x
both grow small. If we examine what happens over an interval
[0
,t
], then we will see that the way that Δ
t
decreases toward 0 should be related to the way
that Δ
x
decreases toward 0 in order for something interesting to happen. We look at the
random walk at time
t
. By time
t
, the random walk will have taken
b
t/
Δ
t
c
steps, where
b
x
c
is the greatest integer less than
x
, i.e., the integer part of
x
. However, each step is only
±
Δ
x
.
We can express this special random walk at time
t
directly in terms of a simple random
walk with steps
±
1 as follows. Let the process we are constructing at time
t
be
X
(
t
) = Δ
x
(
Y
1
+
Y
2
+
···
+
Y
b
t/
Δ
t
c
) = Δ
xS
b
t/
Δ
t
c
,
(1)
where
{
Y
k
:
k
≥
1
}
is the sequence of steps in a simple random walk, i.e., a sequence of i.i.d.
random variables with
P
(
Y
k
= 1) =
P
(
Y
k
=

1) = 1
/
2 for all
k
≥
1
,
(2)
and
{
S
k
:
k
≥
1
}
is the random walk itself, i.e., the associated sequence of partial sums deﬁned
by
S
k
≡
Y
1
+
Y
2
+
···
+
Y
k
,
k
≥
1
,
(3)
where we understand
S
0
≡
0. (See (10.1) in the book.)
It suﬃces to consider the mean and variance of the random variable
X
(
t
) deﬁned in (1).
First, the mean is
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 01/16/2012 for the course IEOR 4106 taught by Professor Whitward during the Spring '11 term at Columbia College.
 Spring '11
 WhitWard

Click to edit the document details