Stochastic Calculus, Spring 2010, 22 January,
Lecture 1
Construction of the Brownian motion
Reading for this lecture (for references see the end of the lecture):
•
[1] pp. 72108
Scaled Random Walks.
Our main object of study in this course will be
stochastic
processes
. Loosely speaking, stochastic process is a collection of random variables
indexed by “time” variable
t
:
X
(
t
)
.
Variable
t
could take values in a discreet set,
for instance in the set of positive integers or it could take values in a continuous
set, for instance in [0
,
∞
)
.
An example of stochastic process is a stock price that
changes in time.
Our main objective for today will be to build a very important stochastic process
called
Brownian Motion
. As you will see, Brownian Motion is a continuoustime
stochastic process, it is usually indexed by [0
,
∞
)
.
To build a Brownian motion
we begin with a simpler stochastic process that is called
Symmetric Random
Walk
(SRW).
To construct a SRW we repeatedly toss a fair coin:
P
(
H
) =
P
(
T
) = 1
/
2
.
We
define the successive outcomes of the tosses by
w
1
, w
2
, w
3
· · · ∈ {
H, T
}
and define
w
=
w
1
w
2
w
3
. . . ,
in other words,
w
is the infinite sequence of tosses and
w
n
is the
outcome of the
n
th
toss. Next, let
X
j
be a random variable defined as
X
j
=
braceleftBigg
1
if
w
j
= H,
−
1
if
w
j
= T.
Define discretetime stochastic process
M
(
n
) in the following way:
M
(0) = 0
, M
(
n
) =
n
summationdisplay
j
=1
X
j
.
The process
M
(
n
) is called a symmetric random walk.
With each step it either
steps up one unit or down one unit, and each of the two possibilities is equally
likely.
Exercise 1.
How many are there sequences of H and T of length
n
? For a fixed
positive integer
n
what is the distribution of
M
(
n
)
,
in particular for a given integer
k
what is
P
(
M
(
n
) =
k
)?
Increments of the SRW.
A random walk has independent increments.
This
means that if we choose nonnegative integers 0 =
n
0
< n
1
<
· · ·
< n
k
,
then the
random variables
M
(
n
1
) =
M
(
n
1
)
−
M
(
n
0
)
, M
(
n
2
)
−
M
(
n
1
)
, . . . , M
(
n
k
)
−
M
(
n
k

1
)
are independent. Each of these random variables
M
(
n
i
+1
)
−
M
(
n
i
) =
n
i
+1
summationdisplay
j
=
n
i
+1
X
j
1
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is called an increment of the random walk. It is the change in the position of the
random walk between times
n
i
and
n
i
+1
.
Increments over nonoverlapping time
intervals are independent because they depend on different coin tosses. Moreover,
E
(
M
(
n
i
+1
)
−
M
(
n
i
)) =
n
i
+1
summationdisplay
j
=
n
i
+1
E
X
j
= 0
and
Var(
M
(
n
i
+1
)
−
M
(
n
i
)) =
n
i
+1
summationdisplay
j
=
n
i
+1
Var
X
j
=
n
i
+1
−
n
i
since Var
X
j
= 1
.
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 Spring '10
 Alexey
 Calculus, Central Limit Theorem, Normal Distribution, Brownian Motion, Probability theory, Stochastic process

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