Chapter 2 Summary
Definition.
We say that
M
0
, M
1
, . . .
is a
martingale with respect to
X
0
, X
1
, . . .
if for any
n
≥
0
(i)
E

M
n

<
∞
, (ii)
M
n
can be determined from the values of
M
0
and
X
m
,
m
≤
n
,
and (iii) for any possible values
x
n
, . . . , x
0
E
(
M
n
+1

M
n

X
n
=
x
n
, X
n

1
=
x
n

1
, . . . X
0
=
x
0
) = 0
Theorem 1.
If
M
n
is martingale then
EM
n
=
EM
0
.
We say that
T
is a
stopping time with respect to
X
n
if the occurrence (or nonoccurrence) of the event “we
stop at time
n
” can be determined by looking at the values of the process up to that time:
X
0
, X
1
, . . . , X
n
.
If, for example,
T
= min
{
n
≥
0 :
X
n
∈
A
}
this is true since
{
T
=
n
}
=
{
X
0
∈
A, . . . X
n

1
∈
A, X
n
∈
A
}
Recall that
a
∧
b
= min
{
a, b
}
. Using this notation
M
T
∧
n
is the process stopped at time
T
.
Theorem 2.
If
M
n
is a martingale and
T
is a stopping time (both with respect to
X
n
) then
M
T
∧
n
is a
martingale.
Theorem 3.
Suppose
M
n
is a martingale and let
T
is a stopping time (both with respect to
X
n
).
If
P
(
T <
∞
) = 1 and there is a constant
K
so that

M
T
∧
n
 ≤
K
for all
n
, then
EM
T
=
EM
0
Review Problems
1. Let
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 DURRETT
 Standard Deviation, Probability theory, ball, Xn

Click to edit the document details