Lecture 11
Some Properties of Brownian Motion
11.1
Reflection principle and the first passage time
Define
M
t
= max
0
≤
u
≤
t
W
u
and recall the first passage time
τ
(
a
) = inf
{
t
≥
0 :
W
t
=
a
}
.
Proposition 11.1
For every
a >
0
,
P
(
M
t
≥
a
) =
P
(
τ
(
a
)
< t
) = 2
P
(
W
t
> a
) = 2

2 Φ(
a/
√
t
)
,
(11.1)
where
Φ(
·
)
denotes the cdf of standard normal distribution.
Note:
The rigorous proof relies on the strong Markov property of
W
, but drawing a picture would
provide a heuristic argument.
Furthermore, Theorem 10.1 implies that
W
*
is also a Brownian
motion with
W
*
t
=
W
τ
(
a
)+
t

W
τ
(
a
)
.
11.2
Behaviors of sample paths
Although being continuous, Brownian paths have “crazy” behaviors.
11.2.1
Zero set
Proposition 11.2
Let
S
=
{
t
≥
0 :
W
t
= 0
}
. Then with probability one,
(
i
)
meas
(
S
) = 0
, where “meas” denotes the Lebesgue measure;
(
ii
)
in every interval
(
t
1
, t
2
)
⊂
IR
+
, there are infinitely many elements of
S
;
(
iii
)
S
is a perfect set, i.e.
S
is closed and dense in itself.
Note:
To show (i), it follows from Fubini’s Theorem that
E
[meas(
S
)] =
E
Z
∞
0
I
{
W
t
=0
}
dt
=
Z
∞
0
P
(
W
t
= 0)
dt
= 0
.
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 Spring '08
 Staff
 Normal Distribution, Brownian Motion, Probability theory, Stochastic process, brownian path

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