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IEOR 4106: Introduction to Operations Research: Stochastic Models
Spring 2009, Professor Whitt
Brownian Motion, Gambler’s Ruin and Martingales
Tuesday, April 28
Sections 10.110.3. Brownian motion
We here discuss the gambler’s ruin problem for Brownian motion, using martingales.
1. Basic Deﬁnitions
Let
B
(
t
) :
t
≥
0
}
be
standard
(
μ
= 0, zerodrift, and
σ
2
= 1, unit variance)
Brownian
motion
(BM).
A stochastic process
{
Y
(
t
) :
t
≥
0
}
is a
martingale
(MG) with respect to another stochas
tic 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.” 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.
But we shall be interested in martingales with respect to BM that are themselves appropriate
functions of BM. It can be shown that 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.
.
A nonnegative random variable
T
is a
stopping time
relative to a 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
.
The
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 Spring '08
 Whitt
 Operations Research

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