Cet ouvrage fait partie de la bibliothèque YouScribe
Obtenez un accès à la bibliothèque pour le lire en ligne
En savoir plus

# Re ected Brownian Bridge area conditioned on its local

De
15 pages
Re ected Brownian Bridge area conditioned on its local time at the origin Philippe Chassaing Guy Louchard y March 28, 2002 Abstract Using some properties of the Airy functions, we analyze the re ected Brownian Bridge area W b conditioned on its local time b at the origin. We give a closed form expression of the Laplace trans- form of W b , a recurrence equation for the moments, leading to an eÆcient computation algorithm and an asymptotic form for the density f(x; b) of W b for x _ 0. 1 Introduction Throughout this paper, the standard Brownian motion (BM) will be denoted by x(t). Other classical BM are the re ected BM: x + (t) := jx(t)j, the Brownian Bridge (BB) on [0; 1]: B(t), the re ected BB on [0; 1]: B + (t), the Brownian Excursion: e(t). The local time of x(t) at a, will be denoted by t + (t; a). Following Janson [9, remark 2.3], we dene W b := R 1 0 B + (t)dt (area of the re ected BB), conditioned on having a local time at the origin equal to b.

• brownian bridge

• airy function

• laplace transform

• conditioned

• function iterates

• bb area

• generalized airy

• standard brownian

Voir plus Voir moins
##### Conditioning

Vous aimerez aussi

on
its
at
hard
eg
)o
):
)a
dis-
m;m
0w
of
lso
la
dt
la
min
for
he
also
of
la
ks.
rst
giv
ts
of
):
Henri
e,
Boulev
erties
of
the
Distribution".
limit
v
Airy
)
functions,
OF
w
)
e

a
(
nalyze
imilar
the
laces,
r
.fr
eected
so
Bro
b
wnian
B
B
the
ridge
description.
area
p
W
conditioned
b
i.e.
conditioned
the
o
[5
n
R
i
m
ts
the
lo
e
cal
(
time
)
b

at
ridge
the
0
o
of
rigin.
e
W
length
Using
r
i
w
v
Janson
e
t
a
ersit
closed
p
form
#
expression
Viola
o
the
f
Bro
t
(
he
l
Laplace
D
trans-
ose
Bruxelles,
b
B-1050
of
W
a
b
Z
,
F
a
(
recurrence
=
e
bt
quation
(
for
he
the
erio
momen
c
ts,
R
l
,
Philipp
to
ariable
an
a
ecien
f
t
forest
computation
m
algorithm
239,
and
description
a
still
n
area
a
presen
symptotic
ed
form
wnian
for
b
t
y
he
l
densit

y
ase
f
ersit
(
b
riomphe,
P
b
They
Abstract
d
f
of
W
excursion
b
0
for
)
x
the
#
w
0.
=
1
1
T
e
tro
o
duction
of
Throughout
"Generalized
this
a
pap
],
e
Guy
r,
e
the
w
standard
0
Bro
)
wnian

m
(
otion
)
(
C
BM)
1
will
t
b
1
e
(
denoted
)
b
where
y
wnian
x
is
(
extended
t
line:
).
b
Other
again
classical
0
BM
)
are
a
t
erv
he
origin
reected
random
BM:
b
x
e
+
t
(
a
t
the
Belgium,
a
=
ith
j
trees
x
506
(
54
t
a
)
A
j
BP
,
W
the
lo
Bro
w
wnian
b
Bridge
a
(
at
BB)
and
o
[9],
n
momen
[
b
0
k
;
=
1]:
e
B
p
(
using
t
inear
),
robing.
the
,
reected
c
B
b
B
y
du
Univ
prop
estigated
Fla
y
;
jolet,
212,
and
B
oblete
+
].
(
obtained
t
Airy
),
istribution
the
distribution
Bro
the
wnian
wnian
Excursion:
area
e
1
(
e
t
t
CP
dt
The
as
l
limit
o
a
c
hard@ulb.ac.b
al
3
time
2
of
m;m
x
,
(
w
t
prop
28,
t
t
call
a
distribution
,
W
w
hassain@iecn.u-nancy
ill
Airy
b
email:c
e
rance,
d
[2
enoted
w
b
a
y
h
t
v
+
W
(

t;
=
a
1
).
e
F
t
d'Informatique,
bt
wing
0
Janson
s
[9
t
,
e
remark
s
2
bs
.3],
hassaing
w
w
e
Z
dene
0
W
(
b
)
:=
e
R
s
1
t
0
B
B
s
+
bs
(
dt
t
t
)
Bro
t
b
(area
B
o
p
f
dically
the
to
reected
whole
BB),
Cedex,
conditioned
euvre

=
D
ving
to
a
1
l
e
o
t
cal
dt
time
using
a
Theorem
t
V
the
aat
origin
t
equal
second
t
The
o
v
b
W
.
ando
Bruxelles,
b
[9
seen
],
s
W
he
b
l
app
w
ears
or
as
path
de
of
limit
random
l
w
a
b
w
m
for
and
m
V
3
o
=
l
2
v
D
with
m;m
s
b
scaling.
p
simple
m
time
,
cal
where
of
D
b
some
h
b
A
p
attempt
m
as
denotes

the
y
Libre
in
email:louc
talk
[
ted
k
A
]
A'2000
oincar
describ
P
in
b
where
t
he
follo
Bridge
Theorem[9,
W
3.3]:
,
Institut

Cartan,
(
CNRS
Bro
nd
E
placemen
W
t
b
for
;
a
g
Reected
hashing
table
y
w
he
ith
wing
m
Th.
places

a
Elie
nd
INRIA,
b
a
p
e
m

1
y
ard
epartemen
Univ
en
iv
are
en
lac
the
es,
ea
des
no
can
ase
the
min
consequence
As
the
(the
in
as
Previously
empt
total
the
In
ha
on
dt
ollo
).
1]:
[0
on
In
x;
of
form
2002
Marc
Louc
the;:::
dx;
Ai
Ai
=0
zt
3.
yK
)=
exp(
j;n
function
is
z;
)o
zW
Ai
Ai
(
z;
)=
]=
)o
(27
value
or
)+
)+
wo
dt
tr
Section
form
ula
MAPLE
of
great
help
in
j
1
e
(
(
A
=
b
ortan
j
o
1
1
j
fs
!
[4],
q
relate
3
(
k
b
j
=
2
1
(

b
f
)
ar
;
1
in
w
which:
1
q
[
r
3
(
y
b
(
)

:=
b
Z
=
+
1
1
5
0
1
x
Theorem
r
0
r
;
!
in
e
:=
bx
prop
x
organized
2
needed
=
to
2
=
j
=
k
u
0
e
(
du:
z
for
)
b
=
0:
(

z
2
)
=

x
z
3
!
=
+
1
1
+
+
=
1
O
X
#
k
in
i
(
!
(
k
smal
(
)
1)
x
k
(
z
0
(3
(34)
k
[11
computations.
a
=
e
2
estigated,
2
a
k
The
:
dev
Here
t
Ai
W
denotes
ts
the
1
c
bz
lassical
2
A
)
iry
u
function
0
complicated
=
z
)]
)
2
:=
2
1
t

symptotic
Z
he
+
(
1
f
0
x
k
Section

b
1
#
3
2
t
2
3
3
+

k
1

=
!
=

=
:=
=
1
x

b

+
z
4
3
4

1
x
=

2
1
K
1
1
4
=
wher
3
is

1.3.
2
the
z
of
3
of
=
)
2
F
3
b
!
x;
:
x
=1
0
K
bf
1
1
=
2
3
e
denotes
f
the
given
Bessel
(35).
function
]
of
a
order
the
1
f
=
1

t
Another
in
expression
Airy
i
oted
s
i
giv
^
en
er
b
s
;
Section
n
giv
i
basic
g
the
h
a
t
m
[10
W
].
i
He
i
obtains
e
Theorem
1
i
3
k
concludes

0
k
u
(
=
b
(
1
)
k
Ai
!
(
G
)
k
Ai
(
u
b
0
)
uz
;
=
with
=
G

n
5
(
w
s
a
)
forms
=
t

densit
s
f
2
x;
=
)
16)
W
3
for
n
#
2
Theorem
X
f
j
x;
=
)
n
x
d
0
j

D
=x
j
p
(
p
s=
"
2)
1
;
4
wher
9
e
4
D
9
j
11
(
4
x
3
)
4
is
3
the
4
p
3
ar
4
ab
3
olic
9
cylinder
4
;k
1
again
4
d
b
as
=
1
3
given
=
by
b
a
16
r
3
e
)
w
7
Y
4
e
3
e
4
quation.
+
In

Section
x
3
=
,

w
:
e
e
pro
2
vide
given
a
(31)
m

ore
is
e
absolute
cien
Theorems
t
rst
r
o
ecurrence
Ai
f
u
orm
.
ula
o
for
k
the
l
computation
,
o
(
f
b
the

momen
#
ts.
f
Our
(
er.

c
1
pap
x
tributions
O
are
b
a
)
closed
wher
f
f
orm
and
e
1
xpression
e
o
in
f
and
t
As
he
[12
Laplace
,
t
]
ransform
nd
(
where
k
l
b
.
X
X
f
R
W
0
b
(
,
)
Theorem
is
the
v
@
the
0
function
b
erties
=1
y
E
n
[
mp
e
eorem
j
ole.
b
pap
X
is
asymptotic
a
of
follo
A
in
function.
2
k
e
w
e
giv
he
a
results
!
in
for
sequel.
momen
e
1
lso
=
the
6
omen
p
of

b
Z
an
z
form
1
the
=
iry
3
In
e
4
b
e
2
e
=
recurrence

2
i
the
2
Section
2
some
and
1.2
pro
the
to
is
ts.
3,
ws:
pla
1.5
zer
the
of
and
1.4
and
1.3
on
other
j;n
enc
curr
and
1.2
and
dt;
cos
Ai(
1)
Thts
forms
of
Ai
t
)=
Ai
(2
t
):
=2
;G
Then
ws
w
wG
wG
wG
ek
):
dt
con
uous
er(see
Ito
t
ds
en
a;
;
)=
a;
1+
G
;
e(
ww
the
ya
(3),
t
0)
;x
dt
0=
1+
=
tt
t
0)
)=
t
Pr
[[
)=0
db
t
Pr
[[
)=
]=
db=t;
BB
standard
ek
wt
)=
0)
0).
tb
=[
(
)]
t
(
u
0)
i
Z
)
(
du

easily
1
+
M
(
d
t;
1
0)
.

2