IEOR 4106, Spring 2011, Professor Whitt
Brownian Motion, Martingales and Stopping Times
Thursday, April 21
1
Martingales
A stochastic process
{
Y
(
t
) :
t
≥
0
}
is a
martingale
(MG) with respect to another stochastic
process
{
Z
(
t
) :
t
≥
0
}
if
E
[
Y
(
t
)

Z
(
u
)
,
0
≤
u
≤
s
] =
Y
(
s
)
for
0
< s < t .
As an extra technical regularity condition, we require that
E
[

Y
(
t
)

]
<
∞
for all
t
as well.
The stochastic process
{
Z
(
t
) :
t
≥
0
}
above is giving relevant information. The segment
{
Z
(
s
) : 0
≤
s
≤
t
}
gives the
history
up to time
t
. Often the information process
Z
is just
the given stochastic process
Y
. Then we just say that
{
Y
(
t
) :
t
≥
0
}
is a
martingale
(MG),
without saying “with respect to.”
Example 1.1
Let
B
(
t
) :
t
≥
0
}
be
standard
(
μ
= 0, zerodrift, and
σ
2
= 1, unit variance)
Brownian motion
(BM). We will use the fact that standard BM
{
B
(
t
) :
t
≥
0
}
is a martingale
with respect to itself. Then we just say that standard BM is a martingale.
The MG property of BM, like almost everything else, is proved by applying the property of
stationary and independent increments
. We show that
E
[
B
(
t
)

B
(
u
)
,
0
≤
u
≤
s
] =
B
(
s
)
for 0
≤
s < t
:
E
[
B
(
t
)

B
(
u
)
,
0
≤
u
≤
s
]
=
E
[
B
(
s
) +
B
(
t
)

B
(
s
)

B
(
u
)
,
0
≤
u
≤
s
]
(add and subtract)
=
E
[
B
(
s
)

B
(
u
)
,
0
≤
u
≤
s
] +
E
[
B
(
t
)

B
(
s
)

B
(
u
)
,
0
≤
u
≤
s
]
(conditional expectation of sum is sum of conditional expectations)
=
B
(
s
) +
E
[
B
(
t
)

B
(
s
)

B
(
u
)
,
0
≤
u
≤
s
]
because
(
E
[
B
(
s
)

B
(
u
)
,
0
≤
u
≤
s
] =
B
(
s
)
we know
B
(
s
), because it is in the condition, so nothing random)
=
B
(
s
) +
E
[
B
(
t
)

B
(
s
)]
(independent increments)
=
B
(
s
) + 0 =
B
(
s
)
(BM has mean 0)
.
But we shall be interested in martingales with respect to BM that are themselves appro
priate functions of BM.
Example 1.2
An example considered below is the stochastic process
{
B
(
t
)
2

t
:
t
≥
0
}
. We
let
Y
(
t
) =
B
(
t
)
2

t
and
Z
(
t
) =
B
(
t
) in the definition above. Thus we say that
{
B
(
t
)
2

t
:
t
≥
0
}
is a MG with respect to BM
{
B
(
t
) :
t
≥
0
}
. However, it can be shown that
{
B
(
t
)
2

t
:
t
≥
0
}
is also a MG with respect to
{
B
(
t
)
2

t
:
t
≥
0
}
.
And similarly for other functions of BM
that we will consider. They too are simply martingales (with respect to their own “internal”
history), but we will simply show that they are martingales with respect to BM.
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2
Stopping Times
A nonnegative random variable
T
is a
stopping time
relative to a continuoustime stochastic
process
{
Z
(
t
) :
t
≥
0
}
if, for any time
t
, the event
{
T
≤
t
}
depends on
Z
(
s
) only for 0
≤
s
≤
t
.
Stopping before time
t
depends only upon the history up to time
t
. The event that a stopping
time
T
is less than or equal to
t
cannot depend on the future of the reference stochastic process
{
Z
(
s
) :
s
≥
0
}
after time
t
.
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 Spring '11
 WhitWard
 Brownian Motion, martingales

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