Stat 150 Stochastic Processes
Spring 2009
Lecture 3: Martingales and hitting probabilities for random walk
Lecturer:
Jim Pitman
•
A sequence of random variables
S
n
,n
∈ {
0
,
1
,
2
,...
}
, is a
martingale
if
1)
E

S
n

<
∞
for each
n
= 0
,
1
,
2
,...
2)
E
(
S
n
+1

(
S
0
,S
1
,...,S
n
)) =
S
n
E.g.,
S
n
=
S
0
+
X
1
+
···
+
X
n
, where
X
i
are diﬀerences,
X
i
=
S
i

S
i

1
. Observe
that (
S
n
) is a martingale if and only
E
(
X
n
+1

S
0
,S
1
,...,S
n
) = 0.
Then say (
X
i
) is a martingale diﬀerence sequence.
Easy example: let
X
i
be independent random variables (of each other, and of
S
0
), where
E
(
X
i
)
≡
0 =
⇒
X
i
are martingale diﬀerences =
⇒
S
n
is a martingale.
•
Some general observations
:
If
S
n
is a martingale, then
E
(
S
n
)
≡
E
(
S
0
) (Constant in
n
).
Proof
: By induction, suppose true for
n
, then
E
(
S
n
+1
) =
E
(
E
(
S
n
+1

(
S
0
,S
1
,...,S
n
))) =
E
(
S
n
) =
E
(
S
0
)
•
Illustration
: Fair coin tossing walk with absorption at barriers 0 and
b
. Process: