Schürger, Klaus
Working Paper
Laplace transforms and suprema of stochastic
processes
Bonn Econ Discussion Papers, No. 10/2002
Provided in Cooperation with:
Bonn Graduate School of Economics (BGSE), University of Bonn
Suggested Citation: Schürger, Klaus (2002) : Laplace transforms and suprema of stochastic
processes, Bonn Econ Discussion Papers, No. 10/2002, University of Bonn, Bonn Graduate
School of Economics (BGSE), Bonn
This Version is available at:
http://hdl.handle.net/10419/22839
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Bonn Econ Discussion Papers
Discussion Pap er 10/2002
Laplace transforms and suprema of stochastic
processes
by
Klaus Sch¨urger
May 2002
Bonn Graduate School of Economics
Department of Economics
University of Bonn
Adenauerallee 24 - 42
D-53113 Bonn
The Bonn Graduate School of Economics is
sponsored by the
LAPLACE TRANSFORMS AND SUPREMA OF
STOCHASTIC PROCESSES
KLAUS SCH
URGER
Department of Statistics, University of Bonn
Abstract.
It is shown that moments of negative order as well
as positive non-integral order of a nonnegative random variable
X
can b e expressed by the Laplace transform of
X
. Applying these
results to certain rst passage times gives explicit formulae for mo-
ments of suprema of Bessel pro cesses as well as strictly stable Levy
processes having no positive jumps.
Key Words
: Laplace transform, Bessel pro cess, Levy pro cess.
0.
Introduction
In the sequel (
B
t
) denotes a
d
-dimensional standard linear Brownian
motion starting at 0
2
IR
d
(denoted BM(
d
)). In Shiryaev (1999,p.251)
a b eautiful trick is used in order to show that if (
B
t
) is a BM(1),
(0.0.1) E
h
sup
0
s
1
j
B
s
j
i
=
p
=
2
:
In fact, the verication of (0.0.1) can be based on the stopping time
(0.0.2)
T
(1) := inf
f
t
0
jj
B
t
j
=1
g
and its Laplace transform
(0.0.3)
'
1
(
t
)=E
exp(
tT
(1))
=
1
cosh(
p
2
t
)
; t
0
:
The latter is easily obtained by applying the optional stopping theorem
to the martingale
cosh(
sB
t
) exp(
s
2
t=
2) (
t
0) for xed
s
0;
see, e.g., Revuz/Yor (1991,p.68) or Rogers/Williams (1994,p.19). Al-
though there is no explicit inversion of the Laplace transform in (0.0.3)
in any particularly useful form, it turns out, however, that (0.0.3) con-
tains enough information in order to yield (0.0.1). In fact, putting
(0.0.4)
M
(
t
)= sup
0
s
t
j
B
s
j
; t
0
;
1
we get by Brownian scaling, for any
t>
0,
P
M
(1)
t
=
P
sup
0
s
1
j
B
s=t
2
j
1
=
P
M
(1
=t
2
)
1
=
P
T
(1)
1
=t
2
=
P
(
T
(1))
1
=
2
t
;
i.e.,
(0.0.5)
M
(1) and (
T
(1))
1
=
2
have the same distribution
which implies
(0.0.6) E[
M
(1)] =
E
h
(
T
(1))
1
=
2
i
:
Next, using the density of a normal distribution with mean 0 and vari-
ance
s
2
=
2we get
(0.0.7)
s
=
2
p
1
Z
0
exp(
(
t=s
)
2
)
dt; s>
0
:
Hence if
X
0 is a random variable having Laplace transform
'
X
we
obtain from (0.0.7) by using the Fubini-Tonelli theorem,
(0.0.8) E
X
1
=
2
=
2
p
1
Z
0
'
X
(
t
2
)
dt:
Applying (0.0.8) to
X
=
T
(1) and taking into account (0.0.6) as well
as (0.0.3) we arrive at
E[
M
(1)] =
2
p
1
Z
0
1
cosh(
p
2
t
)
dt:
Using the substitution
u
=exp(
p
2
t
) we end up with (0.0.1).
In the sequel we rst extend (0.0.8) in two dierent ways (see The-
orems 1.1 and 1.2 in the next section). Using the same pattern of
pro of as b efore allows us to obtain results similar to (0.0.1) for Bessel
pro cesses as well as for a certain class of Levy pro cesses.
1.
Calculation of Moments via Laplace Transforms
We rst derive an extension of (0.0.8). In order to achieve this it is
natural to startwiththeidentity
(1
=
)=
1
Z
0
u
1
=
1
exp(
u
)
du; >
0
:
2