Copyright
c
2009 by Karl Sigman
1
Introduction to Martingales in discrete time
Martingales are stochastic processes that are meant to capture the notion of a fair game in
the context of gambling. In a fair game, each gamble on average, regardless of the past gam
bles, yields no profit or loss. But the reader should not think that martingales are used just
for gambling; they pop up naturally in numerous applications of stochastic modeling.
They
have enough structure to allow for strong general results while also allowing for dependencies
among variables. Thus they deserve the kind of attention that Markov chains do. Gambling,
however, supplies us with insight and intuition through which a great deal of the theory can be
understood.
1.1
Basic definitions and examples
Definition 1.1
A stochastic process
X
=
{
X
n
:
n
≥
0
}
is called a martingale (MG) if
C1:
E
(

X
n

)
<
∞
, n
≥
0
, and
C2:
E
(
X
n
+1

X
0
, . . . , X
n
) =
X
n
, n
≥
0
.
Notice that property C2 can equivalently be stated as
E
(
X
n
+1

X
n

X
0
, . . . , X
n
) = 0
, n
≥
0
.
(1)
In the context of gambling, by letting
X
n
denote your
total fortune
after the
n
th
gamble, this
then captures the notion of a fair game in that on each gamble, regardless of the outcome of
past gambles, your expected change in fortune is 0; on average you neither win or lose any
money.
Taking expected values in C2 yields
E
(
X
n
+1
) =
E
(
X
n
)
, n
≥
0, and we conclude that
E
(
X
n
) =
E
(
X
0
)
, n
≥
0
,
for any MG;
At any time
n
, your expected fortune is the same as it was initially.
For notational simplicity, we shall let
G
n
=
σ
{
X
0
, . . . , X
n
}
denote all the events determined
by the rvs
X
0
, . . . , X
n
, and refer to it as the
information
determined by
X
up to and including
time
n
. Note that
G
n
⊂ G
n
+1
, n
≥
0; information increases as time
n
increases.
Then the martingale property C2 can be expressed nicely as
E
(
X
n
+1
G
n
) =
X
n
, n
≥
0
.
A very important fact is the following which we will make great use of throughout our study
of martingales:
1
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Proposition 1.1
Suppose that
X
is a stochastic process satisfying C1.
Let
G
n
=
σ
{
X
0
, . . . , X
n
}
, n
≥
0
. Suppose that
F
n
=
σ
{
U
0
, . . . , U
n
}
, n
≥
0
is information for
some other stochastic process such that it contains the information of
X
:
G
n
⊂ F
n
, n
≥
0
.
Then if
E
(
X
n
+1
F
n
) =
X
n
, n
≥
0
, then in fact
E
(
X
n
+1
G
n
) =
X
n
, n
≥
0
, so
X
is a MG.
G
n
⊂ F
n
implies that
F
n
also determines
X
0
, . . . , X
n
, but may also determine other things
as well. So the above Proposition allows us to verify condition C2 by using more information
than is necessary. In many instances, this helps us verify C2 in a much simpler way than would
be the case if we directly used
G
n
.
Proof :
E
(
X
n
+1
G
n
)
=
E
(
E
(
X
n
+1
F
n
))
G
n
)
=
E
(
X
n
G
n
)
=
X
n
.
The first equality follows since
G
n
⊂ F
n
; we can always condition first on more information.
The second equality follows from the assumption that
E
(
X
n
+1
F
n
) =
X
n
, and the third from
the fact that
X
n
is determined by
G
n
.
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 Summer '10
 KarlSigma
 Stochastic process, Markov chain, Random walk, Xn, Dominated convergence theorem

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